Average Error: 10.1 → 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 r1048224 = x;
        double r1048225 = y;
        double r1048226 = z;
        double r1048227 = r1048226 - r1048224;
        double r1048228 = r1048225 * r1048227;
        double r1048229 = r1048224 + r1048228;
        double r1048230 = r1048229 / r1048226;
        return r1048230;
}
↓
double f(double x, double y, double z) {
        double r1048231 = 1.0;
        double r1048232 = y;
        double r1048233 = r1048231 - r1048232;
        double r1048234 = x;
        double r1048235 = z;
        double r1048236 = r1048234 / r1048235;
        double r1048237 = fma(r1048233, r1048236, r1048232);
        return r1048237;
}

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 2019362 +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))