Average Error: 10.1 → 0.0
Time: 4.2s
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 r838337 = x;
        double r838338 = y;
        double r838339 = z;
        double r838340 = r838339 - r838337;
        double r838341 = r838338 * r838340;
        double r838342 = r838337 + r838341;
        double r838343 = r838342 / r838339;
        return r838343;
}
↓
double f(double x, double y, double z) {
        double r838344 = 1.0;
        double r838345 = y;
        double r838346 = r838344 - r838345;
        double r838347 = x;
        double r838348 = z;
        double r838349 = r838347 / r838348;
        double r838350 = fma(r838346, r838349, r838345);
        return r838350;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.1
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.1

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

  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 2019353 +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))