Average Error: 10.2 → 0.0
Time: 2.9s
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 r879703 = x;
        double r879704 = y;
        double r879705 = z;
        double r879706 = r879705 - r879703;
        double r879707 = r879704 * r879706;
        double r879708 = r879703 + r879707;
        double r879709 = r879708 / r879705;
        return r879709;
}
↓
double f(double x, double y, double z) {
        double r879710 = 1.0;
        double r879711 = y;
        double r879712 = r879710 - r879711;
        double r879713 = x;
        double r879714 = z;
        double r879715 = r879713 / r879714;
        double r879716 = fma(r879712, r879715, r879711);
        return r879716;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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. 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 2020057 +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))