Average Error: 0.1 → 0.1
Time: 10.0s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)
double f(double x, double y, double z) {
        double r333478 = x;
        double r333479 = y;
        double r333480 = 0.5;
        double r333481 = r333479 + r333480;
        double r333482 = log(r333479);
        double r333483 = r333481 * r333482;
        double r333484 = r333478 - r333483;
        double r333485 = r333484 + r333479;
        double r333486 = z;
        double r333487 = r333485 - r333486;
        return r333487;
}
↓
double f(double x, double y, double z) {
        double r333488 = x;
        double r333489 = y;
        double r333490 = log(r333489);
        double r333491 = 0.5;
        double r333492 = r333489 + r333491;
        double r333493 = z;
        double r333494 = fma(r333490, r333492, r333493);
        double r333495 = r333494 - r333489;
        double r333496 = r333488 - r333495;
        return r333496;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

  2. Simplified0.1

    \[\leadsto \color{blue}{x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)}\]

  3. Final simplification0.1

    \[\leadsto x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))