Average Error: 10.5 → 0.0
Time: 19.3s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, -y, y + \frac{x}{z}\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, -y, y + \frac{x}{z}\right)
double f(double x, double y, double z) {
        double r33751747 = x;
        double r33751748 = y;
        double r33751749 = z;
        double r33751750 = r33751749 - r33751747;
        double r33751751 = r33751748 * r33751750;
        double r33751752 = r33751747 + r33751751;
        double r33751753 = r33751752 / r33751749;
        return r33751753;
}
↓
double f(double x, double y, double z) {
        double r33751754 = x;
        double r33751755 = z;
        double r33751756 = r33751754 / r33751755;
        double r33751757 = y;
        double r33751758 = -r33751757;
        double r33751759 = r33751757 + r33751756;
        double r33751760 = fma(r33751756, r33751758, r33751759);
        return r33751760;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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

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

  5. Final simplification0.0

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

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))