Average Error: 9.4 → 0.0
Time: 13.8s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, -y, y + \frac{x}{z}\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, -y, y + \frac{x}{z}\right)
double f(double x, double y, double z) {
        double r26167233 = x;
        double r26167234 = y;
        double r26167235 = z;
        double r26167236 = r26167235 - r26167233;
        double r26167237 = r26167234 * r26167236;
        double r26167238 = r26167233 + r26167237;
        double r26167239 = r26167238 / r26167235;
        return r26167239;
}
↓
double f(double x, double y, double z) {
        double r26167240 = x;
        double r26167241 = z;
        double r26167242 = r26167240 / r26167241;
        double r26167243 = y;
        double r26167244 = -r26167243;
        double r26167245 = r26167243 + r26167242;
        double r26167246 = fma(r26167242, r26167244, r26167245);
        return r26167246;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.4
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.4

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

  2. Simplified9.4

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

  3. Taylor expanded around 0 3.3

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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))