Average Error: 10.5 → 0.0
Time: 13.4s
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 r32482642 = x;
        double r32482643 = y;
        double r32482644 = z;
        double r32482645 = r32482644 - r32482642;
        double r32482646 = r32482643 * r32482645;
        double r32482647 = r32482642 + r32482646;
        double r32482648 = r32482647 / r32482644;
        return r32482648;
}
↓
double f(double x, double y, double z) {
        double r32482649 = x;
        double r32482650 = z;
        double r32482651 = r32482649 / r32482650;
        double r32482652 = y;
        double r32482653 = -r32482652;
        double r32482654 = fma(r32482651, r32482653, r32482651);
        double r32482655 = r32482654 + r32482652;
        return r32482655;
}

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 3.9

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

  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 2019174 +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))