Average Error: 0.1 → 0.1
Time: 10.7s
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 r70477 = x;
        double r70478 = y;
        double r70479 = log(r70478);
        double r70480 = r70477 * r70479;
        double r70481 = z;
        double r70482 = r70480 + r70481;
        double r70483 = t;
        double r70484 = r70482 + r70483;
        double r70485 = a;
        double r70486 = r70484 + r70485;
        double r70487 = b;
        double r70488 = 0.5;
        double r70489 = r70487 - r70488;
        double r70490 = c;
        double r70491 = log(r70490);
        double r70492 = r70489 * r70491;
        double r70493 = r70486 + r70492;
        double r70494 = i;
        double r70495 = r70478 * r70494;
        double r70496 = r70493 + r70495;
        return r70496;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r70497 = i;
        double r70498 = y;
        double r70499 = c;
        double r70500 = log(r70499);
        double r70501 = b;
        double r70502 = 0.5;
        double r70503 = r70501 - r70502;
        double r70504 = r70500 * r70503;
        double r70505 = x;
        double r70506 = log(r70498);
        double r70507 = z;
        double r70508 = fma(r70505, r70506, r70507);
        double r70509 = t;
        double r70510 = a;
        double r70511 = r70509 + r70510;
        double r70512 = r70508 + r70511;
        double r70513 = r70504 + r70512;
        double r70514 = fma(r70497, r70498, r70513);
        return r70514;
}

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 2020047 +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)))