Average Error: 0.2 → 0.0
Time: 3.3s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)
double f(double x, double y, double z) {
        double r5260 = 4.0;
        double r5261 = x;
        double r5262 = y;
        double r5263 = r5261 - r5262;
        double r5264 = z;
        double r5265 = 0.5;
        double r5266 = r5264 * r5265;
        double r5267 = r5263 - r5266;
        double r5268 = r5260 * r5267;
        double r5269 = r5268 / r5264;
        return r5269;
}
↓
double f(double x, double y, double z) {
        double r5270 = 4.0;
        double r5271 = x;
        double r5272 = z;
        double r5273 = r5271 / r5272;
        double r5274 = y;
        double r5275 = r5274 / r5272;
        double r5276 = r5273 - r5275;
        double r5277 = 2.0;
        double r5278 = -r5277;
        double r5279 = fma(r5270, r5276, r5278);
        return r5279;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.2

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

  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. Using strategy rm

  5. Applied div-sub0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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))