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 r838739 = x;
        double r838740 = y;
        double r838741 = r838739 * r838740;
        double r838742 = z;
        double r838743 = r838741 / r838742;
        return r838743;
}
↓
double f(double x, double y, double z) {
        double r838744 = x;
        double r838745 = cbrt(r838744);
        double r838746 = r838745 * r838745;
        double r838747 = z;
        double r838748 = cbrt(r838747);
        double r838749 = r838748 * r838748;
        double r838750 = y;
        double r838751 = cbrt(r838750);
        double r838752 = r838751 * r838751;
        double r838753 = r838749 / r838752;
        double r838754 = r838746 / r838753;
        double r838755 = r838748 / r838751;
        double r838756 = r838745 / r838755;
        double r838757 = r838754 * r838756;
        return r838757;
}

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 +o rules:numerics
(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))