Average Error: 10.4 → 0.7
Time: 24.5s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\left(y + \frac{x}{z}\right) - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{x}{\frac{z}{\sqrt[3]{y}}}\right) + \frac{x}{\frac{z}{\sqrt[3]{y}}} \cdot \left(\left(-\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \sqrt[3]{y} \cdot \sqrt[3]{y}\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\left(y + \frac{x}{z}\right) - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{x}{\frac{z}{\sqrt[3]{y}}}\right) + \frac{x}{\frac{z}{\sqrt[3]{y}}} \cdot \left(\left(-\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \sqrt[3]{y} \cdot \sqrt[3]{y}\right)
double f(double x, double y, double z) {
        double r551430 = x;
        double r551431 = y;
        double r551432 = z;
        double r551433 = r551432 - r551430;
        double r551434 = r551431 * r551433;
        double r551435 = r551430 + r551434;
        double r551436 = r551435 / r551432;
        return r551436;
}
↓
double f(double x, double y, double z) {
        double r551437 = y;
        double r551438 = x;
        double r551439 = z;
        double r551440 = r551438 / r551439;
        double r551441 = r551437 + r551440;
        double r551442 = cbrt(r551437);
        double r551443 = r551442 * r551442;
        double r551444 = r551439 / r551442;
        double r551445 = r551438 / r551444;
        double r551446 = r551443 * r551445;
        double r551447 = r551441 - r551446;
        double r551448 = -r551443;
        double r551449 = r551448 + r551443;
        double r551450 = r551445 * r551449;
        double r551451 = r551447 + r551450;
        return r551451;
}

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
Target0.0
Herbie0.7
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.4

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

  2. Simplified10.4

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}}\]

  3. Taylor expanded around 0 3.7

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

  5. Applied associate-/l*2.9

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

  7. Applied add-cube-cbrt3.0

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

  8. Applied *-un-lft-identity3.0

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

  9. Applied times-frac3.0

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

  10. Applied add-sqr-sqrt32.9

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

  11. Applied times-frac31.9

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

  12. Applied add-sqr-sqrt48.1

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

  13. Applied prod-diff48.1

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

  14. Simplified31.9

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

  15. Simplified0.7

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

  16. Final simplification0.7

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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