Average Error: 10.4 → 0.7
Time: 24.4s
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 r448127 = x;
        double r448128 = y;
        double r448129 = z;
        double r448130 = r448129 - r448127;
        double r448131 = r448128 * r448130;
        double r448132 = r448127 + r448131;
        double r448133 = r448132 / r448129;
        return r448133;
}
↓
double f(double x, double y, double z) {
        double r448134 = y;
        double r448135 = x;
        double r448136 = z;
        double r448137 = r448135 / r448136;
        double r448138 = r448134 + r448137;
        double r448139 = cbrt(r448134);
        double r448140 = r448139 * r448139;
        double r448141 = r448136 / r448139;
        double r448142 = r448135 / r448141;
        double r448143 = r448140 * r448142;
        double r448144 = r448138 - r448143;
        double r448145 = -r448140;
        double r448146 = r448145 + r448140;
        double r448147 = r448142 * r448146;
        double r448148 = r448144 + r448147;
        return r448148;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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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))