Average Error: 0.2 → 0.0
Time: 1.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r272732 = 1.0;
        double r272733 = 4.0;
        double r272734 = x;
        double r272735 = y;
        double r272736 = 0.75;
        double r272737 = r272735 * r272736;
        double r272738 = r272734 + r272737;
        double r272739 = z;
        double r272740 = r272738 - r272739;
        double r272741 = r272733 * r272740;
        double r272742 = r272741 / r272735;
        double r272743 = r272732 + r272742;
        return r272743;
}
↓
double f(double x, double y, double z) {
        double r272744 = 4.0;
        double r272745 = x;
        double r272746 = y;
        double r272747 = r272745 / r272746;
        double r272748 = z;
        double r272749 = r272748 / r272746;
        double r272750 = r272744 * r272749;
        double r272751 = r272744 - r272750;
        double r272752 = fma(r272744, r272747, r272751);
        return r272752;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +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)))