Average Error: 0.1 → 0.1
Time: 11.7s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)\]
\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
↓
\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r77486 = x;
        double r77487 = y;
        double r77488 = log(r77487);
        double r77489 = r77486 * r77488;
        double r77490 = z;
        double r77491 = r77489 + r77490;
        double r77492 = t;
        double r77493 = r77491 + r77492;
        double r77494 = a;
        double r77495 = r77493 + r77494;
        double r77496 = b;
        double r77497 = 0.5;
        double r77498 = r77496 - r77497;
        double r77499 = c;
        double r77500 = log(r77499);
        double r77501 = r77498 * r77500;
        double r77502 = r77495 + r77501;
        double r77503 = i;
        double r77504 = r77487 * r77503;
        double r77505 = r77502 + r77504;
        return r77505;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r77506 = i;
        double r77507 = y;
        double r77508 = c;
        double r77509 = log(r77508);
        double r77510 = b;
        double r77511 = 0.5;
        double r77512 = r77510 - r77511;
        double r77513 = r77509 * r77512;
        double r77514 = x;
        double r77515 = log(r77507);
        double r77516 = z;
        double r77517 = fma(r77514, r77515, r77516);
        double r77518 = t;
        double r77519 = a;
        double r77520 = r77518 + r77519;
        double r77521 = r77517 + r77520;
        double r77522 = r77513 + r77521;
        double r77523 = fma(r77506, r77507, r77522);
        return r77523;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)}\right)\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))