Average Error: 9.2 → 0.0
Time: 17.6s
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 r26066450 = x;
        double r26066451 = y;
        double r26066452 = z;
        double r26066453 = r26066452 - r26066450;
        double r26066454 = r26066451 * r26066453;
        double r26066455 = r26066450 + r26066454;
        double r26066456 = r26066455 / r26066452;
        return r26066456;
}
↓
double f(double x, double y, double z) {
        double r26066457 = x;
        double r26066458 = z;
        double r26066459 = r26066457 / r26066458;
        double r26066460 = y;
        double r26066461 = -r26066460;
        double r26066462 = fma(r26066459, r26066461, r26066459);
        double r26066463 = r26066462 + r26066460;
        return r26066463;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.2
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.2

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

  2. Simplified9.1

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

  3. Taylor expanded around 0 2.9

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

  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 2019158 +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))