Average Error: 10.4 → 0.0
Time: 3.0s
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 r704440 = x;
        double r704441 = y;
        double r704442 = z;
        double r704443 = r704442 - r704440;
        double r704444 = r704441 * r704443;
        double r704445 = r704440 + r704444;
        double r704446 = r704445 / r704442;
        return r704446;
}
↓
double f(double x, double y, double z) {
        double r704447 = 1.0;
        double r704448 = y;
        double r704449 = r704447 - r704448;
        double r704450 = x;
        double r704451 = z;
        double r704452 = r704450 / r704451;
        double r704453 = fma(r704449, r704452, r704448);
        return r704453;
}

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 2020039 +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))