Average Error: 10.2 → 0.0
Time: 13.2s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) + \frac{x}{z} \cdot \left(-y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) + \frac{x}{z} \cdot \left(-y\right)
double f(double x, double y, double z) {
        double r710163 = x;
        double r710164 = y;
        double r710165 = z;
        double r710166 = r710165 - r710163;
        double r710167 = r710164 * r710166;
        double r710168 = r710163 + r710167;
        double r710169 = r710168 / r710165;
        return r710169;
}
↓
double f(double x, double y, double z) {
        double r710170 = x;
        double r710171 = z;
        double r710172 = r710170 / r710171;
        double r710173 = y;
        double r710174 = r710172 + r710173;
        double r710175 = -r710173;
        double r710176 = r710172 * r710175;
        double r710177 = r710174 + r710176;
        return r710177;
}

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

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

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 3.5

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

  5. Applied sub-neg3.5

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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))