Average Error: 0.1 → 0.0
Time: 13.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r363699 = 1.0;
        double r363700 = 4.0;
        double r363701 = x;
        double r363702 = y;
        double r363703 = 0.25;
        double r363704 = r363702 * r363703;
        double r363705 = r363701 + r363704;
        double r363706 = z;
        double r363707 = r363705 - r363706;
        double r363708 = r363700 * r363707;
        double r363709 = r363708 / r363702;
        double r363710 = r363699 + r363709;
        return r363710;
}
↓
double f(double x, double y, double z) {
        double r363711 = 4.0;
        double r363712 = x;
        double r363713 = z;
        double r363714 = r363712 - r363713;
        double r363715 = y;
        double r363716 = r363714 / r363715;
        double r363717 = 2.0;
        double r363718 = fma(r363711, r363716, r363717);
        return r363718;
}

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

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

  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

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

  5. Final simplification0.0

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

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)))