Average Error: 0.1 → 0.0
Time: 1.3s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)
double f(double x, double y, double z) {
        double r427200 = 4.0;
        double r427201 = x;
        double r427202 = y;
        double r427203 = r427201 - r427202;
        double r427204 = z;
        double r427205 = 0.5;
        double r427206 = r427204 * r427205;
        double r427207 = r427203 - r427206;
        double r427208 = r427200 * r427207;
        double r427209 = r427208 / r427204;
        return r427209;
}
↓
double f(double x, double y, double z) {
        double r427210 = 4.0;
        double r427211 = x;
        double r427212 = y;
        double r427213 = r427211 - r427212;
        double r427214 = z;
        double r427215 = r427213 / r427214;
        double r427216 = 2.0;
        double r427217 = -r427216;
        double r427218 = fma(r427210, r427215, r427217);
        return r427218;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.0
Herbie0.0
\[4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right)\]

Derivation

  1. Initial program 0.1

    \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - \left(4 \cdot \frac{y}{z} + 2\right)}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z))))

  (/ (* 4 (- (- x y) (* z 0.5))) z))