Average Error: 0.1 → 0.1
Time: 18.1s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\mathsf{fma}\left(1, y + x, -\mathsf{fma}\left(y + 0.5, \log y, z\right)\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\mathsf{fma}\left(1, y + x, -\mathsf{fma}\left(y + 0.5, \log y, z\right)\right)
double f(double x, double y, double z) {
        double r292069 = x;
        double r292070 = y;
        double r292071 = 0.5;
        double r292072 = r292070 + r292071;
        double r292073 = log(r292070);
        double r292074 = r292072 * r292073;
        double r292075 = r292069 - r292074;
        double r292076 = r292075 + r292070;
        double r292077 = z;
        double r292078 = r292076 - r292077;
        return r292078;
}
↓
double f(double x, double y, double z) {
        double r292079 = 1.0;
        double r292080 = y;
        double r292081 = x;
        double r292082 = r292080 + r292081;
        double r292083 = 0.5;
        double r292084 = r292080 + r292083;
        double r292085 = log(r292080);
        double r292086 = z;
        double r292087 = fma(r292084, r292085, r292086);
        double r292088 = -r292087;
        double r292089 = fma(r292079, r292082, r292088);
        return r292089;
}

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

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

  4. Applied *-un-lft-identity0.1

    \[\leadsto \color{blue}{1 \cdot \left(y + x\right)} - \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

  5. Applied fma-neg0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, y + x, -\mathsf{fma}\left(y + 0.5, \log y, z\right)\right)}\]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(1, y + x, -\mathsf{fma}\left(y + 0.5, \log y, z\right)\right)\]

Reproduce

herbie shell --seed 2019208 +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))