Average Error: 6.5 → 2.1
Time: 6.8s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\]
\frac{x \cdot y}{z}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}
double f(double x, double y, double z) {
        double r820782 = x;
        double r820783 = y;
        double r820784 = r820782 * r820783;
        double r820785 = z;
        double r820786 = r820784 / r820785;
        return r820786;
}
↓
double f(double x, double y, double z) {
        double r820787 = x;
        double r820788 = cbrt(r820787);
        double r820789 = r820788 * r820788;
        double r820790 = z;
        double r820791 = cbrt(r820790);
        double r820792 = r820791 * r820791;
        double r820793 = y;
        double r820794 = cbrt(r820793);
        double r820795 = r820794 * r820794;
        double r820796 = r820792 / r820795;
        double r820797 = r820789 / r820796;
        double r820798 = r820791 / r820794;
        double r820799 = r820788 / r820798;
        double r820800 = r820797 * r820799;
        return r820800;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.5
Target6.0
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Initial program 6.5

    \[\frac{x \cdot y}{z}\]
  2. Using strategy rm

  3. Applied associate-/l*6.0

    \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt6.8

    \[\leadsto \frac{x}{\frac{z}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

  6. Applied add-cube-cbrt7.0

    \[\leadsto \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

  7. Applied times-frac7.0

    \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}\]

  8. Applied add-cube-cbrt7.1

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\]

  9. Applied times-frac2.1

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}\]

  10. Final simplification2.1

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))