Average Error: 0.1 → 0.9
Time: 24.7s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \mathsf{fma}\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}\right)}^{3}} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \mathsf{fma}\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}\right)}^{3}} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -y\right)
double f(double x, double y, double z) {
        double r351098 = x;
        double r351099 = y;
        double r351100 = 0.5;
        double r351101 = r351099 + r351100;
        double r351102 = log(r351099);
        double r351103 = r351101 * r351102;
        double r351104 = r351098 - r351103;
        double r351105 = r351104 + r351099;
        double r351106 = z;
        double r351107 = r351105 - r351106;
        return r351107;
}
↓
double f(double x, double y, double z) {
        double r351108 = x;
        double r351109 = y;
        double r351110 = log(r351109);
        double r351111 = 0.5;
        double r351112 = r351109 + r351111;
        double r351113 = z;
        double r351114 = fma(r351110, r351112, r351113);
        double r351115 = cbrt(r351114);
        double r351116 = 3.0;
        double r351117 = pow(r351115, r351116);
        double r351118 = cbrt(r351117);
        double r351119 = r351118 * r351115;
        double r351120 = -r351109;
        double r351121 = fma(r351119, r351115, r351120);
        double r351122 = r351108 - r351121;
        return r351122;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.9
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

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 add-cube-cbrt0.9

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

  5. Applied fma-neg0.9

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

  7. Applied add-cbrt-cube0.9

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

  8. Simplified0.9

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

  9. Final simplification0.9

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

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))