Average Error: 10.5 → 0.0
Time: 15.1s
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 r50324067 = x;
        double r50324068 = y;
        double r50324069 = z;
        double r50324070 = r50324069 - r50324067;
        double r50324071 = r50324068 * r50324070;
        double r50324072 = r50324067 + r50324071;
        double r50324073 = r50324072 / r50324069;
        return r50324073;
}
↓
double f(double x, double y, double z) {
        double r50324074 = x;
        double r50324075 = z;
        double r50324076 = r50324074 / r50324075;
        double r50324077 = -r50324076;
        double r50324078 = y;
        double r50324079 = r50324078 + r50324076;
        double r50324080 = fma(r50324077, r50324078, r50324079);
        return r50324080;
}

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

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

  3. Taylor expanded around 0 3.9

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 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))