Average Error: 10.3 → 0.0
Time: 27.8s
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 r24089933 = x;
        double r24089934 = y;
        double r24089935 = z;
        double r24089936 = r24089935 - r24089933;
        double r24089937 = r24089934 * r24089936;
        double r24089938 = r24089933 + r24089937;
        double r24089939 = r24089938 / r24089935;
        return r24089939;
}
↓
double f(double x, double y, double z) {
        double r24089940 = x;
        double r24089941 = z;
        double r24089942 = r24089940 / r24089941;
        double r24089943 = y;
        double r24089944 = -r24089943;
        double r24089945 = fma(r24089942, r24089944, r24089942);
        double r24089946 = r24089945 + r24089943;
        return r24089946;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.3

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

  2. Simplified10.3

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

  3. Taylor expanded around 0 3.3

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

  4. Simplified0.0

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

  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))