Average Error: 10.2 → 0.0
Time: 12.7s
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 r950801 = x;
        double r950802 = y;
        double r950803 = z;
        double r950804 = r950803 - r950801;
        double r950805 = r950802 * r950804;
        double r950806 = r950801 + r950805;
        double r950807 = r950806 / r950803;
        return r950807;
}
↓
double f(double x, double y, double z) {
        double r950808 = x;
        double r950809 = z;
        double r950810 = r950808 / r950809;
        double r950811 = y;
        double r950812 = r950810 + r950811;
        double r950813 = -r950808;
        double r950814 = r950813 / r950809;
        double r950815 = r950814 * r950811;
        double r950816 = r950812 + r950815;
        return r950816;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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. Simplified3.2

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

  8. Applied associate-/r/0.0

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

  9. 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))