Average Error: 0.1 → 0.0
Time: 11.1s
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 r362997 = 1.0;
        double r362998 = 4.0;
        double r362999 = x;
        double r363000 = y;
        double r363001 = 0.25;
        double r363002 = r363000 * r363001;
        double r363003 = r362999 + r363002;
        double r363004 = z;
        double r363005 = r363003 - r363004;
        double r363006 = r362998 * r363005;
        double r363007 = r363006 / r363000;
        double r363008 = r362997 + r363007;
        return r363008;
}
↓
double f(double x, double y, double z) {
        double r363009 = x;
        double r363010 = z;
        double r363011 = r363009 - r363010;
        double r363012 = y;
        double r363013 = r363011 / r363012;
        double r363014 = 4.0;
        double r363015 = 2.0;
        double r363016 = fma(r363013, r363014, r363015);
        return r363016;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 +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)))