Average Error: 9.8 → 0.2
Time: 10.0s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}
double f(double x, double y, double z) {
        double r39840251 = x;
        double r39840252 = y;
        double r39840253 = z;
        double r39840254 = r39840253 - r39840251;
        double r39840255 = r39840252 * r39840254;
        double r39840256 = r39840251 + r39840255;
        double r39840257 = r39840256 / r39840253;
        return r39840257;
}
↓
double f(double x, double y, double z) {
        double r39840258 = x;
        double r39840259 = z;
        double r39840260 = r39840258 / r39840259;
        double r39840261 = y;
        double r39840262 = r39840260 + r39840261;
        double r39840263 = cbrt(r39840261);
        double r39840264 = cbrt(r39840259);
        double r39840265 = r39840263 / r39840264;
        double r39840266 = r39840258 * r39840265;
        double r39840267 = r39840265 * r39840266;
        double r39840268 = r39840267 * r39840265;
        double r39840269 = r39840262 - r39840268;
        return r39840269;
}

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

Original9.8
Target0.0
Herbie0.2
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.8

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

  2. Taylor expanded around 0 3.4

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

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

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

  5. Applied times-frac3.3

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

  6. Simplified3.3

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

  8. Applied add-cube-cbrt3.5

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

  9. Applied add-cube-cbrt3.5

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

  10. Applied times-frac3.5

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

  11. Applied associate-*r*0.7

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

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

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