Average Error: 10.2 → 0.0
Time: 12.5s
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 r810967 = x;
        double r810968 = y;
        double r810969 = z;
        double r810970 = r810969 - r810967;
        double r810971 = r810968 * r810970;
        double r810972 = r810967 + r810971;
        double r810973 = r810972 / r810969;
        return r810973;
}
↓
double f(double x, double y, double z) {
        double r810974 = x;
        double r810975 = z;
        double r810976 = r810974 / r810975;
        double r810977 = y;
        double r810978 = r810976 + r810977;
        double r810979 = -r810977;
        double r810980 = r810976 * r810979;
        double r810981 = r810978 + r810980;
        return r810981;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

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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))