Average Error: 0.1 → 0.1
Time: 18.2s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x - \mathsf{fma}\left(\log y, y + 0.5, -y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x - \mathsf{fma}\left(\log y, y + 0.5, -y\right)\right) - z
double f(double x, double y, double z) {
        double r216541 = x;
        double r216542 = y;
        double r216543 = 0.5;
        double r216544 = r216542 + r216543;
        double r216545 = log(r216542);
        double r216546 = r216544 * r216545;
        double r216547 = r216541 - r216546;
        double r216548 = r216547 + r216542;
        double r216549 = z;
        double r216550 = r216548 - r216549;
        return r216550;
}
↓
double f(double x, double y, double z) {
        double r216551 = x;
        double r216552 = y;
        double r216553 = log(r216552);
        double r216554 = 0.5;
        double r216555 = r216552 + r216554;
        double r216556 = -r216552;
        double r216557 = fma(r216553, r216555, r216556);
        double r216558 = r216551 - r216557;
        double r216559 = z;
        double r216560 = r216558 - r216559;
        return r216560;
}

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. Using strategy rm

  3. Applied associate-+l-0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"

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

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