Average Error: 10.5 → 0.0
Time: 16.0s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\frac{x}{z} + \left(\frac{x}{z} \cdot \left(-y\right) + y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\frac{x}{z} + \left(\frac{x}{z} \cdot \left(-y\right) + y\right)
double f(double x, double y, double z) {
        double r31365524 = x;
        double r31365525 = y;
        double r31365526 = z;
        double r31365527 = r31365526 - r31365524;
        double r31365528 = r31365525 * r31365527;
        double r31365529 = r31365524 + r31365528;
        double r31365530 = r31365529 / r31365526;
        return r31365530;
}
↓
double f(double x, double y, double z) {
        double r31365531 = x;
        double r31365532 = z;
        double r31365533 = r31365531 / r31365532;
        double r31365534 = y;
        double r31365535 = -r31365534;
        double r31365536 = r31365533 * r31365535;
        double r31365537 = r31365536 + r31365534;
        double r31365538 = r31365533 + r31365537;
        return r31365538;
}

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

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

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 3.5

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

  4. Simplified0.0

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

  6. Applied fma-udef0.0

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

  7. Applied associate-+r+0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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))