Average Error: 10.3 → 0.0
Time: 1.5s
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 r752121 = x;
        double r752122 = y;
        double r752123 = z;
        double r752124 = r752123 - r752121;
        double r752125 = r752122 * r752124;
        double r752126 = r752121 + r752125;
        double r752127 = r752126 / r752123;
        return r752127;
}
↓
double f(double x, double y, double z) {
        double r752128 = 1.0;
        double r752129 = y;
        double r752130 = r752128 - r752129;
        double r752131 = x;
        double r752132 = z;
        double r752133 = r752131 / r752132;
        double r752134 = fma(r752130, r752133, r752129);
        return r752134;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.3

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

  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 2020064 +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))