Average Error: 9.8 → 0.0
Time: 19.2s
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 r29213315 = x;
        double r29213316 = y;
        double r29213317 = z;
        double r29213318 = r29213317 - r29213315;
        double r29213319 = r29213316 * r29213318;
        double r29213320 = r29213315 + r29213319;
        double r29213321 = r29213320 / r29213317;
        return r29213321;
}
↓
double f(double x, double y, double z) {
        double r29213322 = x;
        double r29213323 = z;
        double r29213324 = r29213322 / r29213323;
        double r29213325 = y;
        double r29213326 = -r29213325;
        double r29213327 = fma(r29213324, r29213326, r29213324);
        double r29213328 = r29213327 + r29213325;
        return r29213328;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.8
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.8

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

  2. Simplified9.8

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

  3. Taylor expanded around 0 3.2

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

  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 2019162 +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))