Average Error: 9.9 → 0.0
Time: 2.5s
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 r675303 = x;
        double r675304 = y;
        double r675305 = z;
        double r675306 = r675305 - r675303;
        double r675307 = r675304 * r675306;
        double r675308 = r675303 + r675307;
        double r675309 = r675308 / r675305;
        return r675309;
}
↓
double f(double x, double y, double z) {
        double r675310 = 1.0;
        double r675311 = y;
        double r675312 = r675310 - r675311;
        double r675313 = x;
        double r675314 = z;
        double r675315 = r675313 / r675314;
        double r675316 = fma(r675312, r675315, r675311);
        return r675316;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.9
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.9

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

  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 2020062 +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))