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Rate is a PERCENT (5.25 = 5.25%). Calculation tool only — no taxes or …"},{"id":"npv-irr","name":"NPV and IRR of a cash-flow series","tags":["finance","calculation","deterministic"],"description":"Net present value at a per-period percent rate and the internal rate of return of up to 30 cash flows (period 0 first; negative = outflow). IRR is returned only when it is unique (exactly one sign change); otherwise the request is refused before payment (no_sign_change / multiple_irr / no_converge). Rate is a PERCENT per period. Calculation tool only — not investment advice. Price 400 µAIC."},{"id":"loan-schedule","name":"Loan repayment summary (annuity or equal principal)","tags":["finance","calculation","deterministic"],"description":"Monthly loan repayment: first and last installment (payment, principal, interest, balance), total interest and total paid, for an annuity (level payment) or equal-principal loan over 1–1200 months. Annual rate is a PERCENT (4.5 = 4.5%, at most 100). Principal at most 9999999.99. Amounts are rounded to 4 decimals. Long terms at high rates are refused with 422 ill_conditioned (rounding error of the annuity recursion grows with (1 + rate/12)^months; amplification above 1e5 is not sold). Fixed rate…"},{"id":"bond-ytm-duration","name":"Bond price, yield to maturity, duration and convexity","tags":["finance","bond","calculation","deterministic"],"description":"Fixed-coupon bullet bond valued on a coupon date (first coupon exactly one period away, so NO accrued interest: clean = dirty price). Give face, coupon_pct, years, m (coupon payments per year: 1, 2, 4 or 12) and EXACTLY ONE of y_pct (yield → price) or price (price → yield, solved by bisection). Yields and rates are PERCENT, nominal annual compounded m times a year (per-period rate = y/m), not an effective annual rate. years × m must be a whole number of coupon periods (m=12: use multiples of 0.…"},{"id":"black-scholes","name":"Black-Scholes price and Greeks (European option)","tags":["finance","options","black-scholes","greeks","calculation","deterministic"],"description":"European call and put under Black-Scholes-Merton: constant volatility and rate, continuous dividend yield q, no early exercise. Inputs: S spot, K strike, T years to expiry, r_pct risk-free rate, sigma_pct volatility, q_pct dividend yield (default 0) — all PERCENT, annualised; r and q are continuously compounded. Returns d1, d2, call, put and the Greeks with the ENGINE'S conventions: delta{call,put} per 1 unit of S; gamma per 1 unit of S²; vega = ∂V/∂σ per 1.00 change of volatility (20% → 120%),…"},{"id":"prob-distribution","name":"Probability distribution: probabilities and quantiles (binomial, Poisson, normal, t, chi-square, F)","tags":["statistics","probability","calculation","deterministic"],"description":"Cumulative / interval probabilities and quantiles. kind with p1/p2: binom (p1 = n trials 1–100000, p2 = success probability 0–1), poisson (p1 = lambda > 0 up to 100000, p2 omitted), normal (p1 = mean, p2 = standard deviation > 0), t (p1 = degrees of freedom, integer 1–1000), chi2 (p1 = degrees of freedom, integer 1–1000), f (p1, p2 = numerator and denominator degrees of freedom, integers 1–1000). query: le = P(X ≤ a); ge = P(X ≥ a); between = P(a ≤ X ≤ b), bounds included; quantile = smallest i…"},{"id":"confidence-interval","name":"Confidence interval for a mean (t) or a proportion (Wald and Wilson), with an optional one-sample test","tags":["statistics","confidence-interval","calculation","deterministic"],"description":"kind=mean: from summary statistics mean, sd (sample standard deviation, n − 1), n (2–1000) and level (0.5 < level ≤ 0.999, up to 4 decimals, default 0.95): standard error, t critical value, margin of error and the interval mean ± t·se; with mu0 also the one-sample t statistic, degrees of freedom and two-sided p. kind=proportion: x successes of n trials (n up to 1e9) at level: p-hat, the Wald interval (clipped to [0,1]; unreliable for small n or extreme p, flagged by small_sample) and the Wilson…"},{"id":"sample-size-power","name":"Statistical power and required sample size (t tests, two proportions, correlation)","tags":["statistics","power-analysis","sample-size","calculation","deterministic"],"description":"Give exactly one of n (get the power at that sample size) or power (get the required sample size). kind: ttest_ind = independent two-sample t test (effect = Cohen d, n = size of group 1, group 2 = ratio × n, ratio 0.1–10, default 1); ttest_one = one-sample or paired t test (effect = d of the differences); prop = two proportions by Cohen h with a normal approximation (p1, p2 in (0,1), p1 ≠ p2, n = size of group 1, group 2 = ratio × n); corr = correlation test by Fisher z (effect = r, 0 < |r| < 1…"},{"id":"welch-ttest","name":"Welch's two-sample t-test (unequal variances)","tags":["statistics","calculation","deterministic"],"description":"Two-sided Welch t-test of two independent samples a and b (2–50 decimals each, comma-separated, sign allowed). Returns t = (mean(a) − mean(b)) / sqrt(sd_a²/n_a + sd_b²/n_b), Welch–Satterthwaite df, the two-sided p, and per group n, mean, sd (n−1), se, median, min, max. Derived values carry 9 significant digits; p is rounded to 6 decimals (a reported p of 0 means p < 5e-7 — the engine's tail p is not reliable below ~1e-10 in relative terms: measured relative error 7e-7 at p=1e-10, 7% at 1e-15; i…"},{"id":"exact-integer-math","name":"Exact integer math: modpow, modinv, crt, binom, egcd","tags":["math","number-theory","calculation","deterministic"],"description":"Exact big-integer arithmetic, results as decimal STRINGS (never rounded). op=modpow: a^exp mod mod (a may be negative; result in [0, mod); 0^0 = 1; exp ≥ 0, mod ≥ 1). op=modinv: inverse of a modulo mod (mod ≥ 2; refused 422 noinv if gcd(a mod mod, mod) ≠ 1). op=crt: Chinese remainder theorem for 1–10 congruences x ≡ remainders[i] (mod moduli[i]) (moduli ≥ 1, not necessarily coprime; result x in [0, lcm) plus the combined modulus; refused 422 nosol if inconsistent). op=binom: C(n, k) for 0 ≤ n, …"},{"id":"ipv4-subnet","name":"IPv4 subnet calculator (CIDR, split, host-count sizing)","tags":["network","ipv4","calculation","deterministic"],"description":"Given one IPv4 address with prefix in strict CIDR form a.b.c.d/p (octets 0-255 without leading zeros, prefix 0-32; no netmask form), returns network, broadcast, first/last usable host, mask, wildcard, usable host count, total addresses, legacy class (from the first octet of the given address) and RFC 1918 private flag, ALL ADDRESSES AS OCTET ARRAYS such as [192,168,1,0] (never dotted strings). Rules: usable hosts = 2^(32-p) - 2 (network and broadcast excluded); /31 = 2 usable hosts (RFC 3021 po…"},{"id":"beam-deflection","name":"Beam reactions, moments, shear and maximum deflection (6 standard cases)","tags":["engineering","structural","calculation","deterministic"],"description":"Linear-elastic Euler-Bernoulli beam of constant EI in SI units only (metres, newtons, pascals, m^4). case: ss_udl = simply supported, uniform load w (N/m); ss_point = simply supported, point load P (N) at a_m from the left support (a_m must be between span_m/1000 and span_m*(1-1/1000)); cant_point = cantilever fixed at x=0, point load P at the free end x=L; cant_udl = cantilever, uniform load w; ff_udl = both ends fully fixed, uniform load w; cont2_udl = two equal spans L (total 2L, supports 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