Average Error: 0.2 → 0.0
Time: 1.8s
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 r266832 = 1.0;
        double r266833 = 4.0;
        double r266834 = x;
        double r266835 = y;
        double r266836 = 0.75;
        double r266837 = r266835 * r266836;
        double r266838 = r266834 + r266837;
        double r266839 = z;
        double r266840 = r266838 - r266839;
        double r266841 = r266833 * r266840;
        double r266842 = r266841 / r266835;
        double r266843 = r266832 + r266842;
        return r266843;
}
↓
double f(double x, double y, double z) {
        double r266844 = 4.0;
        double r266845 = x;
        double r266846 = y;
        double r266847 = r266845 / r266846;
        double r266848 = z;
        double r266849 = r266848 / r266846;
        double r266850 = r266844 * r266849;
        double r266851 = r266844 - r266850;
        double r266852 = fma(r266844, r266847, r266851);
        return r266852;
}

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 2020064 +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)))