Average Error: 10.4 → 0.9
Time: 2.9s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\left(\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + 1 \cdot \frac{x}{z}\right) - x\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\left(\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + 1 \cdot \frac{x}{z}\right) - x
double f(double x, double y, double z) {
        double r618538 = x;
        double r618539 = y;
        double r618540 = z;
        double r618541 = r618539 - r618540;
        double r618542 = 1.0;
        double r618543 = r618541 + r618542;
        double r618544 = r618538 * r618543;
        double r618545 = r618544 / r618540;
        return r618545;
}
↓
double f(double x, double y, double z) {
        double r618546 = x;
        double r618547 = y;
        double r618548 = cbrt(r618547);
        double r618549 = r618548 * r618548;
        double r618550 = z;
        double r618551 = cbrt(r618550);
        double r618552 = r618551 * r618551;
        double r618553 = r618549 / r618552;
        double r618554 = r618546 * r618553;
        double r618555 = r618548 / r618551;
        double r618556 = r618554 * r618555;
        double r618557 = 1.0;
        double r618558 = r618546 / r618550;
        double r618559 = r618557 * r618558;
        double r618560 = r618556 + r618559;
        double r618561 = r618560 - r618546;
        return r618561;
}

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

Original10.4
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.4

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

  2. Taylor expanded around 0 3.6

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

  4. Applied *-un-lft-identity3.6

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

  5. Applied times-frac3.3

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

  6. Simplified3.3

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

  8. Applied add-cube-cbrt3.6

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

  9. Applied add-cube-cbrt3.7

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

  10. Applied times-frac3.7

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

  11. Applied associate-*r*0.9

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

  12. Final simplification0.9

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

Reproduce

herbie shell --seed 2020081 
(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))