Average Error: 0.1 → 0.1
Time: 5.9s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z
double f(double x, double y, double z) {
        double r540701 = x;
        double r540702 = y;
        double r540703 = 0.5;
        double r540704 = r540702 + r540703;
        double r540705 = log(r540702);
        double r540706 = r540704 * r540705;
        double r540707 = r540701 - r540706;
        double r540708 = r540707 + r540702;
        double r540709 = z;
        double r540710 = r540708 - r540709;
        return r540710;
}
↓
double f(double x, double y, double z) {
        double r540711 = x;
        double r540712 = y;
        double r540713 = 0.5;
        double r540714 = r540712 + r540713;
        double r540715 = -r540714;
        double r540716 = log(r540712);
        double r540717 = fma(r540715, r540716, r540712);
        double r540718 = r540711 + r540717;
        double r540719 = z;
        double r540720 = r540718 - r540719;
        return r540720;
}

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

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019362 +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))