Average Error: 0.1 → 0.0
Time: 1.8s
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 r869404 = 4.0;
        double r869405 = x;
        double r869406 = y;
        double r869407 = r869405 - r869406;
        double r869408 = z;
        double r869409 = 0.5;
        double r869410 = r869408 * r869409;
        double r869411 = r869407 - r869410;
        double r869412 = r869404 * r869411;
        double r869413 = r869412 / r869408;
        return r869413;
}
↓
double f(double x, double y, double z) {
        double r869414 = 4.0;
        double r869415 = x;
        double r869416 = y;
        double r869417 = r869415 - r869416;
        double r869418 = z;
        double r869419 = r869417 / r869418;
        double r869420 = 2.0;
        double r869421 = -r869420;
        double r869422 = fma(r869414, r869419, r869421);
        return r869422;
}

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 2020047 +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))