Average Error: 0.2 → 0.0
Time: 1.3s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z}, -\mathsf{fma}\left(4, \frac{y}{z}, 2\right)\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z}, -\mathsf{fma}\left(4, \frac{y}{z}, 2\right)\right)
double f(double x, double y, double z) {
        double r2333170 = 4.0;
        double r2333171 = x;
        double r2333172 = y;
        double r2333173 = r2333171 - r2333172;
        double r2333174 = z;
        double r2333175 = 0.5;
        double r2333176 = r2333174 * r2333175;
        double r2333177 = r2333173 - r2333176;
        double r2333178 = r2333170 * r2333177;
        double r2333179 = r2333178 / r2333174;
        return r2333179;
}
↓
double f(double x, double y, double z) {
        double r2333180 = 4.0;
        double r2333181 = x;
        double r2333182 = z;
        double r2333183 = r2333181 / r2333182;
        double r2333184 = y;
        double r2333185 = r2333184 / r2333182;
        double r2333186 = 2.0;
        double r2333187 = fma(r2333180, r2333185, r2333186);
        double r2333188 = -r2333187;
        double r2333189 = fma(r2333180, r2333183, r2333188);
        return r2333189;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - \mathsf{fma}\left(0.5, z, y\right)\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +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))