Average Error: 10.4 → 0.0
Time: 3.2s
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 r884231 = x;
        double r884232 = y;
        double r884233 = z;
        double r884234 = r884233 - r884231;
        double r884235 = r884232 * r884234;
        double r884236 = r884231 + r884235;
        double r884237 = r884236 / r884233;
        return r884237;
}
↓
double f(double x, double y, double z) {
        double r884238 = 1.0;
        double r884239 = y;
        double r884240 = r884238 - r884239;
        double r884241 = x;
        double r884242 = z;
        double r884243 = r884241 / r884242;
        double r884244 = fma(r884240, r884243, r884239);
        return r884244;
}

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 2020083 +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))