Average Error: 0.2 → 0.0
Time: 12.3s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4.0, 0.75 + \frac{x - z}{y}, 1.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4.0, 0.75 + \frac{x - z}{y}, 1.0\right)
double f(double x, double y, double z) {
        double r13996732 = 1.0;
        double r13996733 = 4.0;
        double r13996734 = x;
        double r13996735 = y;
        double r13996736 = 0.75;
        double r13996737 = r13996735 * r13996736;
        double r13996738 = r13996734 + r13996737;
        double r13996739 = z;
        double r13996740 = r13996738 - r13996739;
        double r13996741 = r13996733 * r13996740;
        double r13996742 = r13996741 / r13996735;
        double r13996743 = r13996732 + r13996742;
        return r13996743;
}
↓
double f(double x, double y, double z) {
        double r13996744 = 4.0;
        double r13996745 = 0.75;
        double r13996746 = x;
        double r13996747 = z;
        double r13996748 = r13996746 - r13996747;
        double r13996749 = y;
        double r13996750 = r13996748 / r13996749;
        double r13996751 = r13996745 + r13996750;
        double r13996752 = 1.0;
        double r13996753 = fma(r13996744, r13996751, r13996752);
        return r13996753;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4.0, 0.75 + \frac{x - z}{y}, 1.0\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4.0, 0.75 + \frac{x - z}{y}, 1.0\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))