Average Error: 0.1 → 0.1
Time: 12.2s
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 r352021 = x;
        double r352022 = y;
        double r352023 = 0.5;
        double r352024 = r352022 + r352023;
        double r352025 = log(r352022);
        double r352026 = r352024 * r352025;
        double r352027 = r352021 - r352026;
        double r352028 = r352027 + r352022;
        double r352029 = z;
        double r352030 = r352028 - r352029;
        return r352030;
}
↓
double f(double x, double y, double z) {
        double r352031 = x;
        double r352032 = y;
        double r352033 = log(r352032);
        double r352034 = 0.5;
        double r352035 = r352032 + r352034;
        double r352036 = z;
        double r352037 = fma(r352033, r352035, r352036);
        double r352038 = r352037 - r352032;
        double r352039 = r352031 - r352038;
        return r352039;
}

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))