Average Error: 10.2 → 0.0
Time: 13.6s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \frac{x}{z} \cdot y\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \frac{x}{z} \cdot y
double f(double x, double y, double z) {
        double r730422 = x;
        double r730423 = y;
        double r730424 = z;
        double r730425 = r730424 - r730422;
        double r730426 = r730423 * r730425;
        double r730427 = r730422 + r730426;
        double r730428 = r730427 / r730424;
        return r730428;
}
↓
double f(double x, double y, double z) {
        double r730429 = x;
        double r730430 = z;
        double r730431 = r730429 / r730430;
        double r730432 = y;
        double r730433 = r730431 + r730432;
        double r730434 = r730431 * r730432;
        double r730435 = r730433 - r730434;
        return r730435;
}

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

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 associate-/l*3.2

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

  7. Applied *-un-lft-identity3.2

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

  8. Applied *-un-lft-identity3.2

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

  9. Applied times-frac3.2

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

  10. Applied *-un-lft-identity3.2

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

  11. Applied times-frac3.2

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

  12. Simplified3.2

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))