Average Error: 10.0 → 0.0
Time: 3.1s
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 r689494 = x;
        double r689495 = y;
        double r689496 = z;
        double r689497 = r689496 - r689494;
        double r689498 = r689495 * r689497;
        double r689499 = r689494 + r689498;
        double r689500 = r689499 / r689496;
        return r689500;
}
↓
double f(double x, double y, double z) {
        double r689501 = 1.0;
        double r689502 = y;
        double r689503 = r689501 - r689502;
        double r689504 = x;
        double r689505 = z;
        double r689506 = r689504 / r689505;
        double r689507 = fma(r689503, r689506, r689502);
        return r689507;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 10.0

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

  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 2020035 +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))