Average Error: 0.1 → 0.1
Time: 4.9s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x - \mathsf{fma}\left(\log y, y + 0.5, z - y\right)\right) + \mathsf{fma}\left(-\left(z - y\right), 1, z - y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x - \mathsf{fma}\left(\log y, y + 0.5, z - y\right)\right) + \mathsf{fma}\left(-\left(z - y\right), 1, z - y\right)
double f(double x, double y, double z) {
        double r347825 = x;
        double r347826 = y;
        double r347827 = 0.5;
        double r347828 = r347826 + r347827;
        double r347829 = log(r347826);
        double r347830 = r347828 * r347829;
        double r347831 = r347825 - r347830;
        double r347832 = r347831 + r347826;
        double r347833 = z;
        double r347834 = r347832 - r347833;
        return r347834;
}
↓
double f(double x, double y, double z) {
        double r347835 = x;
        double r347836 = y;
        double r347837 = log(r347836);
        double r347838 = 0.5;
        double r347839 = r347836 + r347838;
        double r347840 = z;
        double r347841 = r347840 - r347836;
        double r347842 = fma(r347837, r347839, r347841);
        double r347843 = r347835 - r347842;
        double r347844 = -r347841;
        double r347845 = 1.0;
        double r347846 = fma(r347844, r347845, r347841);
        double r347847 = r347843 + r347846;
        return r347847;
}

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. Using strategy rm

  4. Applied fma-udef0.1

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

  5. Applied associate--l+0.1

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

  6. Applied associate--r+0.1

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied add-sqr-sqrt36.6

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

  10. Applied prod-diff36.6

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

  11. Simplified0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +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))