Average Error: 10.2 → 0.0
Time: 2.1s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)
double f(double x, double y, double z) {
        double r741643 = x;
        double r741644 = y;
        double r741645 = z;
        double r741646 = r741645 - r741643;
        double r741647 = r741644 * r741646;
        double r741648 = r741643 + r741647;
        double r741649 = r741648 / r741645;
        return r741649;
}
↓
double f(double x, double y, double z) {
        double r741650 = 1.0;
        double r741651 = y;
        double r741652 = r741650 - r741651;
        double r741653 = x;
        double r741654 = z;
        double r741655 = r741653 / r741654;
        double r741656 = fma(r741652, r741655, r741651);
        return r741656;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))