Average Error: 10.2 → 0.1
Time: 13.1s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \frac{\frac{x}{z}}{\frac{1}{y}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \frac{\frac{x}{z}}{\frac{1}{y}}
double f(double x, double y, double z) {
        double r653164 = x;
        double r653165 = y;
        double r653166 = z;
        double r653167 = r653166 - r653164;
        double r653168 = r653165 * r653167;
        double r653169 = r653164 + r653168;
        double r653170 = r653169 / r653166;
        return r653170;
}
↓
double f(double x, double y, double z) {
        double r653171 = x;
        double r653172 = z;
        double r653173 = r653171 / r653172;
        double r653174 = y;
        double r653175 = r653173 + r653174;
        double r653176 = 1.0;
        double r653177 = r653176 / r653174;
        double r653178 = r653173 / r653177;
        double r653179 = r653175 - r653178;
        return r653179;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original10.2
Target0.0
Herbie0.1
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

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

  2. Taylor expanded around 0 3.5

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

  4. Applied associate-/l*3.2

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

  6. Applied div-inv3.2

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 
(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))