Average Error: 0.1 → 0.1
Time: 12.3s
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 r358536 = x;
        double r358537 = y;
        double r358538 = 0.5;
        double r358539 = r358537 + r358538;
        double r358540 = log(r358537);
        double r358541 = r358539 * r358540;
        double r358542 = r358536 - r358541;
        double r358543 = r358542 + r358537;
        double r358544 = z;
        double r358545 = r358543 - r358544;
        return r358545;
}
↓
double f(double x, double y, double z) {
        double r358546 = x;
        double r358547 = y;
        double r358548 = log(r358547);
        double r358549 = 0.5;
        double r358550 = r358547 + r358549;
        double r358551 = z;
        double r358552 = fma(r358548, r358550, r358551);
        double r358553 = r358552 - r358547;
        double r358554 = r358546 - r358553;
        return r358554;
}

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 2020001 +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))