Average Error: 0.1 → 0.1
Time: 48.3s
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 r361415 = x;
        double r361416 = y;
        double r361417 = 0.5;
        double r361418 = r361416 + r361417;
        double r361419 = log(r361416);
        double r361420 = r361418 * r361419;
        double r361421 = r361415 - r361420;
        double r361422 = r361421 + r361416;
        double r361423 = z;
        double r361424 = r361422 - r361423;
        return r361424;
}
↓
double f(double x, double y, double z) {
        double r361425 = x;
        double r361426 = y;
        double r361427 = log(r361426);
        double r361428 = -r361427;
        double r361429 = 0.5;
        double r361430 = r361426 + r361429;
        double r361431 = fma(r361428, r361430, r361426);
        double r361432 = r361425 + r361431;
        double r361433 = z;
        double r361434 = r361432 - r361433;
        return r361434;
}

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 sub-neg0.1

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

  4. Applied associate-+l+0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019322 +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))