Average Error: 10.5 → 0.1
Time: 12.9s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(y + \frac{-\frac{x}{z}}{\frac{1}{y}}\right) + \frac{x}{z}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(y + \frac{-\frac{x}{z}}{\frac{1}{y}}\right) + \frac{x}{z}
double f(double x, double y, double z) {
        double r511263 = x;
        double r511264 = y;
        double r511265 = z;
        double r511266 = r511265 - r511263;
        double r511267 = r511264 * r511266;
        double r511268 = r511263 + r511267;
        double r511269 = r511268 / r511265;
        return r511269;
}
↓
double f(double x, double y, double z) {
        double r511270 = y;
        double r511271 = x;
        double r511272 = z;
        double r511273 = r511271 / r511272;
        double r511274 = -r511273;
        double r511275 = 1.0;
        double r511276 = r511275 / r511270;
        double r511277 = r511274 / r511276;
        double r511278 = r511270 + r511277;
        double r511279 = r511278 + r511273;
        return r511279;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original10.5
Target0.0
Herbie0.1
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.5

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

  2. Simplified10.5

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}}\]

  3. Taylor expanded around 0 3.4

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

  4. Simplified0.0

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

  6. Applied fma-udef0.0

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

  7. Simplified3.1

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

  9. Applied div-inv3.1

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

  10. Applied associate-/r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))