Average Error: 0.1 → 0.1
Time: 5.9s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]
\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 r359749 = x;
        double r359750 = y;
        double r359751 = 0.5;
        double r359752 = r359750 + r359751;
        double r359753 = log(r359750);
        double r359754 = r359752 * r359753;
        double r359755 = r359749 - r359754;
        double r359756 = r359755 + r359750;
        double r359757 = z;
        double r359758 = r359756 - r359757;
        return r359758;
}
↓
double f(double x, double y, double z) {
        double r359759 = x;
        double r359760 = y;
        double r359761 = log(r359760);
        double r359762 = 0.5;
        double r359763 = r359760 + r359762;
        double r359764 = z;
        double r359765 = fma(r359761, r359763, r359764);
        double r359766 = r359765 - r359760;
        double r359767 = r359759 - r359766;
        return r359767;
}

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 2020057 +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))