Average Error: 10.4 → 0.0
Time: 2.6s
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 r757517 = x;
        double r757518 = y;
        double r757519 = z;
        double r757520 = r757519 - r757517;
        double r757521 = r757518 * r757520;
        double r757522 = r757517 + r757521;
        double r757523 = r757522 / r757519;
        return r757523;
}
↓
double f(double x, double y, double z) {
        double r757524 = 1.0;
        double r757525 = y;
        double r757526 = r757524 - r757525;
        double r757527 = x;
        double r757528 = z;
        double r757529 = r757527 / r757528;
        double r757530 = fma(r757526, r757529, r757525);
        return r757530;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.4

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

  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 2020046 +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))