Average Error: 9.6 → 0.0
Time: 14.1s
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 r35927478 = x;
        double r35927479 = y;
        double r35927480 = z;
        double r35927481 = r35927480 - r35927478;
        double r35927482 = r35927479 * r35927481;
        double r35927483 = r35927478 + r35927482;
        double r35927484 = r35927483 / r35927480;
        return r35927484;
}
↓
double f(double x, double y, double z) {
        double r35927485 = x;
        double r35927486 = z;
        double r35927487 = r35927485 / r35927486;
        double r35927488 = y;
        double r35927489 = -r35927488;
        double r35927490 = fma(r35927487, r35927489, r35927487);
        double r35927491 = r35927490 + r35927488;
        return r35927491;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.6
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.6

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

  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 2019168 +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))