Average Error: 0.2 → 0.0
Time: 2.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r227233 = 1.0;
        double r227234 = 4.0;
        double r227235 = x;
        double r227236 = y;
        double r227237 = 0.75;
        double r227238 = r227236 * r227237;
        double r227239 = r227235 + r227238;
        double r227240 = z;
        double r227241 = r227239 - r227240;
        double r227242 = r227234 * r227241;
        double r227243 = r227242 / r227236;
        double r227244 = r227233 + r227243;
        return r227244;
}
↓
double f(double x, double y, double z) {
        double r227245 = 4.0;
        double r227246 = x;
        double r227247 = y;
        double r227248 = r227246 / r227247;
        double r227249 = z;
        double r227250 = r227249 / r227247;
        double r227251 = r227245 * r227250;
        double r227252 = r227245 - r227251;
        double r227253 = fma(r227245, r227248, r227252);
        return r227253;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

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, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))