Average Error: 9.9 → 0.0
Time: 2.3s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)
double f(double x, double y, double z) {
        double r648745 = x;
        double r648746 = y;
        double r648747 = z;
        double r648748 = r648747 - r648745;
        double r648749 = r648746 * r648748;
        double r648750 = r648745 + r648749;
        double r648751 = r648750 / r648747;
        return r648751;
}
↓
double f(double x, double y, double z) {
        double r648752 = 1.0;
        double r648753 = y;
        double r648754 = r648752 - r648753;
        double r648755 = x;
        double r648756 = z;
        double r648757 = r648755 / r648756;
        double r648758 = fma(r648754, r648757, r648753);
        return r648758;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.9
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.9

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

  2. Simplified0.0

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

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)\]

Reproduce

herbie shell --seed 2020062 +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))