Average Error: 0.1 → 0.0
Time: 16.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r283181 = 1.0;
        double r283182 = 4.0;
        double r283183 = x;
        double r283184 = y;
        double r283185 = 0.25;
        double r283186 = r283184 * r283185;
        double r283187 = r283183 + r283186;
        double r283188 = z;
        double r283189 = r283187 - r283188;
        double r283190 = r283182 * r283189;
        double r283191 = r283190 / r283184;
        double r283192 = r283181 + r283191;
        return r283192;
}
↓
double f(double x, double y, double z) {
        double r283193 = x;
        double r283194 = z;
        double r283195 = r283193 - r283194;
        double r283196 = y;
        double r283197 = r283195 / r283196;
        double r283198 = 4.0;
        double r283199 = 2.0;
        double r283200 = fma(r283197, r283198, r283199);
        return r283200;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))