Average Error: 0.1 → 0.0
Time: 1.4s
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 r272813 = 1.0;
        double r272814 = 4.0;
        double r272815 = x;
        double r272816 = y;
        double r272817 = 0.75;
        double r272818 = r272816 * r272817;
        double r272819 = r272815 + r272818;
        double r272820 = z;
        double r272821 = r272819 - r272820;
        double r272822 = r272814 * r272821;
        double r272823 = r272822 / r272816;
        double r272824 = r272813 + r272823;
        return r272824;
}
↓
double f(double x, double y, double z) {
        double r272825 = 4.0;
        double r272826 = x;
        double r272827 = y;
        double r272828 = r272826 / r272827;
        double r272829 = z;
        double r272830 = r272829 / r272827;
        double r272831 = r272825 * r272830;
        double r272832 = r272825 - r272831;
        double r272833 = fma(r272825, r272828, r272832);
        return r272833;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  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 2019353 +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)))