Average Error: 0.1 → 0.1
Time: 28.4s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(\left(x - \mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{y}\right), y + 0.5, \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(\left(x - \mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{y}\right), y + 0.5, \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z
double f(double x, double y, double z) {
        double r358930 = x;
        double r358931 = y;
        double r358932 = 0.5;
        double r358933 = r358931 + r358932;
        double r358934 = log(r358931);
        double r358935 = r358933 * r358934;
        double r358936 = r358930 - r358935;
        double r358937 = r358936 + r358931;
        double r358938 = z;
        double r358939 = r358937 - r358938;
        return r358939;
}
↓
double f(double x, double y, double z) {
        double r358940 = x;
        double r358941 = 2.0;
        double r358942 = y;
        double r358943 = cbrt(r358942);
        double r358944 = log(r358943);
        double r358945 = r358941 * r358944;
        double r358946 = 0.5;
        double r358947 = r358942 + r358946;
        double r358948 = r358947 * r358944;
        double r358949 = fma(r358945, r358947, r358948);
        double r358950 = r358940 - r358949;
        double r358951 = r358950 + r358942;
        double r358952 = z;
        double r358953 = r358951 - r358952;
        return r358953;
}

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 add-cube-cbrt0.1

    \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right) + y\right) - z\]

  4. Applied log-prod0.2

    \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z\]

  5. Applied distribute-lft-in0.2

    \[\leadsto \left(\left(x - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z\]

  6. Simplified0.2

    \[\leadsto \left(\left(x - \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)} + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z\]
  7. Using strategy rm

  8. Applied fma-def0.1

    \[\leadsto \left(\left(x - \color{blue}{\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{y}\right), y + 0.5, \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z\]

  9. Final simplification0.1

    \[\leadsto \left(\left(x - \mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{y}\right), y + 0.5, \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z\]

Reproduce

herbie shell --seed 2020045 +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))