Average Error: 10.2 → 1.0
Time: 19.0s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}
double f(double x, double y, double z) {
        double r528668 = x;
        double r528669 = y;
        double r528670 = z;
        double r528671 = r528670 - r528668;
        double r528672 = r528669 * r528671;
        double r528673 = r528668 + r528672;
        double r528674 = r528673 / r528670;
        return r528674;
}
↓
double f(double x, double y, double z) {
        double r528675 = x;
        double r528676 = z;
        double r528677 = r528675 / r528676;
        double r528678 = y;
        double r528679 = r528677 + r528678;
        double r528680 = cbrt(r528676);
        double r528681 = r528680 * r528680;
        double r528682 = r528675 / r528681;
        double r528683 = r528678 / r528680;
        double r528684 = r528682 * r528683;
        double r528685 = r528679 - r528684;
        return r528685;
}

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
Target0.0
Herbie1.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

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

  2. Simplified10.2

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

  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 add-cube-cbrt3.8

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

  6. Applied times-frac1.0

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

  7. Final simplification1.0

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

Reproduce

herbie shell --seed 2019303 +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))