Average Error: 10.2 → 1.4
Time: 3.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\frac{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\frac{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}
double f(double x, double y, double z) {
        double r593645 = x;
        double r593646 = y;
        double r593647 = z;
        double r593648 = r593646 - r593647;
        double r593649 = 1.0;
        double r593650 = r593648 + r593649;
        double r593651 = r593645 * r593650;
        double r593652 = r593651 / r593647;
        return r593652;
}
↓
double f(double x, double y, double z) {
        double r593653 = y;
        double r593654 = z;
        double r593655 = r593653 - r593654;
        double r593656 = 1.0;
        double r593657 = r593655 + r593656;
        double r593658 = cbrt(r593657);
        double r593659 = r593658 * r593658;
        double r593660 = cbrt(r593654);
        double r593661 = r593660 * r593660;
        double r593662 = r593659 / r593661;
        double r593663 = x;
        double r593664 = r593660 / r593658;
        double r593665 = r593663 / r593664;
        double r593666 = r593662 * r593665;
        return r593666;
}

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

Original10.2
Target0.6
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.2

    \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
  2. Using strategy rm

  3. Applied associate-/l*3.1

    \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt3.9

    \[\leadsto \frac{x}{\frac{z}{\color{blue}{\left(\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}\right) \cdot \sqrt[3]{\left(y - z\right) + 1}}}}\]

  6. Applied add-cube-cbrt3.8

    \[\leadsto \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}\right) \cdot \sqrt[3]{\left(y - z\right) + 1}}}\]

  7. Applied times-frac3.8

    \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}}\]

  8. Applied *-un-lft-identity3.8

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}\]

  9. Applied times-frac1.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}}\]

  10. Simplified1.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}\]

  11. Final simplification1.4

    \[\leadsto \frac{\sqrt[3]{\left(y - z\right) + 1} \cdot \sqrt[3]{\left(y - z\right) + 1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{\left(y - z\right) + 1}}}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))