Average Error: 10.5 → 0.0
Time: 16.3s
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 r486735 = x;
        double r486736 = y;
        double r486737 = z;
        double r486738 = r486737 - r486735;
        double r486739 = r486736 * r486738;
        double r486740 = r486735 + r486739;
        double r486741 = r486740 / r486737;
        return r486741;
}
↓
double f(double x, double y, double z) {
        double r486742 = x;
        double r486743 = z;
        double r486744 = r486742 / r486743;
        double r486745 = y;
        double r486746 = r486744 + r486745;
        double r486747 = r486744 * r486745;
        double r486748 = r486746 - r486747;
        return r486748;
}

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

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

  5. Applied pow13.5

    \[\leadsto \color{blue}{{\left(\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}\right)}^{1}}\]
  6. Using strategy rm

  7. Applied associate-/l*3.1

    \[\leadsto {\left(\left(\frac{x}{z} + y\right) - \color{blue}{\frac{x}{\frac{z}{y}}}\right)}^{1}\]
  8. Using strategy rm

  9. Applied associate-/r/0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019199 +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))