Average Error: 0.1 → 0.0
Time: 12.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r222260 = 1.0;
        double r222261 = 4.0;
        double r222262 = x;
        double r222263 = y;
        double r222264 = 0.25;
        double r222265 = r222263 * r222264;
        double r222266 = r222262 + r222265;
        double r222267 = z;
        double r222268 = r222266 - r222267;
        double r222269 = r222261 * r222268;
        double r222270 = r222269 / r222263;
        double r222271 = r222260 + r222270;
        return r222271;
}
↓
double f(double x, double y, double z) {
        double r222272 = x;
        double r222273 = y;
        double r222274 = r222272 / r222273;
        double r222275 = z;
        double r222276 = r222275 / r222273;
        double r222277 = r222274 - r222276;
        double r222278 = 4.0;
        double r222279 = 2.0;
        double r222280 = fma(r222277, r222278, r222279);
        return r222280;
}

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

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

  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. Using strategy rm

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))