Average Error: 10.1 → 0.1
Time: 11.5s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[y - \frac{\frac{x}{z}}{\frac{1}{y - 1}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
y - \frac{\frac{x}{z}}{\frac{1}{y - 1}}
double f(double x, double y, double z) {
        double r644099 = x;
        double r644100 = y;
        double r644101 = z;
        double r644102 = r644101 - r644099;
        double r644103 = r644100 * r644102;
        double r644104 = r644099 + r644103;
        double r644105 = r644104 / r644101;
        return r644105;
}
↓
double f(double x, double y, double z) {
        double r644106 = y;
        double r644107 = x;
        double r644108 = z;
        double r644109 = r644107 / r644108;
        double r644110 = 1.0;
        double r644111 = r644106 - r644110;
        double r644112 = r644110 / r644111;
        double r644113 = r644109 / r644112;
        double r644114 = r644106 - r644113;
        return r644114;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

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

Derivation

  1. Initial program 10.1

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

  2. Taylor expanded around 0 3.6

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

  3. Simplified3.6

    \[\leadsto \color{blue}{y - \frac{y \cdot x - x}{z}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity3.6

    \[\leadsto y - \frac{y \cdot x - \color{blue}{1 \cdot x}}{z}\]

  6. Applied distribute-rgt-out--3.6

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

  7. Applied associate-/l*3.0

    \[\leadsto y - \color{blue}{\frac{x}{\frac{z}{y - 1}}}\]
  8. Using strategy rm

  9. Applied div-inv3.0

    \[\leadsto y - \frac{x}{\color{blue}{z \cdot \frac{1}{y - 1}}}\]

  10. Applied associate-/r*0.1

    \[\leadsto y - \color{blue}{\frac{\frac{x}{z}}{\frac{1}{y - 1}}}\]

  11. Final simplification0.1

    \[\leadsto y - \frac{\frac{x}{z}}{\frac{1}{y - 1}}\]

Reproduce

herbie shell --seed 2019350 
(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))