Average Error: 10.2 → 0.0
Time: 18.2s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(1, \frac{x}{z} + y, \frac{x}{z} \cdot \left(-y\right)\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(1, \frac{x}{z} + y, \frac{x}{z} \cdot \left(-y\right)\right)
double f(double x, double y, double z) {
        double r609115 = x;
        double r609116 = y;
        double r609117 = z;
        double r609118 = r609117 - r609115;
        double r609119 = r609116 * r609118;
        double r609120 = r609115 + r609119;
        double r609121 = r609120 / r609117;
        return r609121;
}
↓
double f(double x, double y, double z) {
        double r609122 = 1.0;
        double r609123 = x;
        double r609124 = z;
        double r609125 = r609123 / r609124;
        double r609126 = y;
        double r609127 = r609125 + r609126;
        double r609128 = -r609126;
        double r609129 = r609125 * r609128;
        double r609130 = fma(r609122, r609127, r609129);
        return r609130;
}

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. Simplified10.2

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

  3. Taylor expanded around 0 3.3

    \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity3.3

    \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{z} + y\right)} - \frac{x \cdot y}{z}\]

  6. Applied fma-neg3.3

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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))