Average Error: 0.2 → 0.0
Time: 1.6s
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 r829223 = 4.0;
        double r829224 = x;
        double r829225 = y;
        double r829226 = r829224 - r829225;
        double r829227 = z;
        double r829228 = 0.5;
        double r829229 = r829227 * r829228;
        double r829230 = r829226 - r829229;
        double r829231 = r829223 * r829230;
        double r829232 = r829231 / r829227;
        return r829232;
}
↓
double f(double x, double y, double z) {
        double r829233 = 4.0;
        double r829234 = x;
        double r829235 = z;
        double r829236 = r829234 / r829235;
        double r829237 = y;
        double r829238 = r829237 / r829235;
        double r829239 = r829236 - r829238;
        double r829240 = 2.0;
        double r829241 = -r829240;
        double r829242 = fma(r829233, r829239, r829241);
        return r829242;
}

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 2020020 +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))