Average Error: 10.4 → 0.0
Time: 19.1s
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 r29180953 = x;
        double r29180954 = y;
        double r29180955 = z;
        double r29180956 = r29180955 - r29180953;
        double r29180957 = r29180954 * r29180956;
        double r29180958 = r29180953 + r29180957;
        double r29180959 = r29180958 / r29180955;
        return r29180959;
}
↓
double f(double x, double y, double z) {
        double r29180960 = x;
        double r29180961 = z;
        double r29180962 = r29180960 / r29180961;
        double r29180963 = y;
        double r29180964 = -r29180963;
        double r29180965 = r29180963 + r29180962;
        double r29180966 = fma(r29180962, r29180964, r29180965);
        return r29180966;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.4

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

  2. Simplified10.4

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

  3. Taylor expanded around 0 3.5

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019169 +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))