Average Error: 10.5 → 0.0
Time: 46.9s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, -y, \frac{x}{z}\right) + y\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, -y, \frac{x}{z}\right) + y
double f(double x, double y, double z) {
        double r34908261 = x;
        double r34908262 = y;
        double r34908263 = z;
        double r34908264 = r34908263 - r34908261;
        double r34908265 = r34908262 * r34908264;
        double r34908266 = r34908261 + r34908265;
        double r34908267 = r34908266 / r34908263;
        return r34908267;
}
↓
double f(double x, double y, double z) {
        double r34908268 = x;
        double r34908269 = z;
        double r34908270 = r34908268 / r34908269;
        double r34908271 = y;
        double r34908272 = -r34908271;
        double r34908273 = fma(r34908270, r34908272, r34908270);
        double r34908274 = r34908273 + r34908271;
        return r34908274;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.5
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.5

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

  2. Simplified10.5

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}}\]

  3. Taylor expanded around 0 3.6

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

  4. Simplified0.0

    \[\leadsto \color{blue}{y + \mathsf{fma}\left(\frac{x}{z}, -y, \frac{x}{z}\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -y, \frac{x}{z}\right) + y\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))