Average Error: 10.2 → 0.0
Time: 2.9s
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 r836401 = x;
        double r836402 = y;
        double r836403 = z;
        double r836404 = r836403 - r836401;
        double r836405 = r836402 * r836404;
        double r836406 = r836401 + r836405;
        double r836407 = r836406 / r836403;
        return r836407;
}
↓
double f(double x, double y, double z) {
        double r836408 = 1.0;
        double r836409 = y;
        double r836410 = r836408 - r836409;
        double r836411 = x;
        double r836412 = z;
        double r836413 = r836411 / r836412;
        double r836414 = fma(r836410, r836413, r836409);
        return r836414;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.1
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 2020025 +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))