Average Error: 0.1 → 0.0
Time: 2.4s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y}, 1 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y}, 1 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r206284 = 1.0;
        double r206285 = 4.0;
        double r206286 = x;
        double r206287 = y;
        double r206288 = 0.25;
        double r206289 = r206287 * r206288;
        double r206290 = r206286 + r206289;
        double r206291 = z;
        double r206292 = r206290 - r206291;
        double r206293 = r206285 * r206292;
        double r206294 = r206293 / r206287;
        double r206295 = r206284 + r206294;
        return r206295;
}
↓
double f(double x, double y, double z) {
        double r206296 = 1.0;
        double r206297 = 4.0;
        double r206298 = x;
        double r206299 = y;
        double r206300 = r206298 / r206299;
        double r206301 = z;
        double r206302 = r206301 / r206299;
        double r206303 = r206297 * r206302;
        double r206304 = r206296 - r206303;
        double r206305 = fma(r206297, r206300, r206304);
        double r206306 = r206296 + r206305;
        return r206306;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020089 +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)))