Average Error: 0.1 → 0.0
Time: 1.1s
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 r888532 = 4.0;
        double r888533 = x;
        double r888534 = y;
        double r888535 = r888533 - r888534;
        double r888536 = z;
        double r888537 = 0.5;
        double r888538 = r888536 * r888537;
        double r888539 = r888535 - r888538;
        double r888540 = r888532 * r888539;
        double r888541 = r888540 / r888536;
        return r888541;
}
↓
double f(double x, double y, double z) {
        double r888542 = 4.0;
        double r888543 = x;
        double r888544 = y;
        double r888545 = r888543 - r888544;
        double r888546 = z;
        double r888547 = r888545 / r888546;
        double r888548 = 2.0;
        double r888549 = -r888548;
        double r888550 = fma(r888542, r888547, r888549);
        return r888550;
}

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 2020033 +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))