Average Error: 10.2 → 1.0
Time: 17.3s
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 r564181 = x;
        double r564182 = y;
        double r564183 = z;
        double r564184 = r564183 - r564181;
        double r564185 = r564182 * r564184;
        double r564186 = r564181 + r564185;
        double r564187 = r564186 / r564183;
        return r564187;
}
↓
double f(double x, double y, double z) {
        double r564188 = x;
        double r564189 = z;
        double r564190 = r564188 / r564189;
        double r564191 = y;
        double r564192 = r564190 + r564191;
        double r564193 = cbrt(r564189);
        double r564194 = r564193 * r564193;
        double r564195 = r564188 / r564194;
        double r564196 = r564191 / r564193;
        double r564197 = r564195 * r564196;
        double r564198 = r564192 - r564197;
        return r564198;
}

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))