Average Error: 9.9 → 0.0
Time: 3.7s
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 r780401 = x;
        double r780402 = y;
        double r780403 = z;
        double r780404 = r780403 - r780401;
        double r780405 = r780402 * r780404;
        double r780406 = r780401 + r780405;
        double r780407 = r780406 / r780403;
        return r780407;
}
↓
double f(double x, double y, double z) {
        double r780408 = 1.0;
        double r780409 = y;
        double r780410 = r780408 - r780409;
        double r780411 = x;
        double r780412 = z;
        double r780413 = r780411 / r780412;
        double r780414 = fma(r780410, r780413, r780409);
        return r780414;
}

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 2020056 +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))