Average Error: 10.2 → 0.0
Time: 3.7s
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 r1503278 = x;
        double r1503279 = y;
        double r1503280 = z;
        double r1503281 = r1503280 - r1503278;
        double r1503282 = r1503279 * r1503281;
        double r1503283 = r1503278 + r1503282;
        double r1503284 = r1503283 / r1503280;
        return r1503284;
}
↓
double f(double x, double y, double z) {
        double r1503285 = 1.0;
        double r1503286 = y;
        double r1503287 = r1503285 - r1503286;
        double r1503288 = x;
        double r1503289 = z;
        double r1503290 = r1503288 / r1503289;
        double r1503291 = fma(r1503287, r1503290, r1503286);
        return r1503291;
}

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 2020034 +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))