Average Error: 0.2 → 0.0
Time: 3.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r251271 = 1.0;
        double r251272 = 4.0;
        double r251273 = x;
        double r251274 = y;
        double r251275 = 0.75;
        double r251276 = r251274 * r251275;
        double r251277 = r251273 + r251276;
        double r251278 = z;
        double r251279 = r251277 - r251278;
        double r251280 = r251272 * r251279;
        double r251281 = r251280 / r251274;
        double r251282 = r251271 + r251281;
        return r251282;
}
↓
double f(double x, double y, double z) {
        double r251283 = 1.0;
        double r251284 = 4.0;
        double r251285 = x;
        double r251286 = y;
        double r251287 = r251285 / r251286;
        double r251288 = 3.0;
        double r251289 = z;
        double r251290 = r251289 / r251286;
        double r251291 = r251284 * r251290;
        double r251292 = r251288 - r251291;
        double r251293 = fma(r251284, r251287, r251292);
        double r251294 = r251283 + r251293;
        return r251294;
}

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

    \[\leadsto 1 + \color{blue}{\left(\left(4 \cdot \frac{x}{y} + 3\right) - 4 \cdot \frac{z}{y}\right)}\]

  3. Simplified0.0

    \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)}\]

  4. Final simplification0.0

    \[\leadsto 1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2020001 +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)))