Average Error: 0.2 → 0.0
Time: 1.3m
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\left(\frac{x}{y} - \frac{z}{y}\right) + 0.75, 4, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(\left(\frac{x}{y} - \frac{z}{y}\right) + 0.75, 4, 1\right)
double f(double x, double y, double z) {
        double r15897610 = 1.0;
        double r15897611 = 4.0;
        double r15897612 = x;
        double r15897613 = y;
        double r15897614 = 0.75;
        double r15897615 = r15897613 * r15897614;
        double r15897616 = r15897612 + r15897615;
        double r15897617 = z;
        double r15897618 = r15897616 - r15897617;
        double r15897619 = r15897611 * r15897618;
        double r15897620 = r15897619 / r15897613;
        double r15897621 = r15897610 + r15897620;
        return r15897621;
}
↓
double f(double x, double y, double z) {
        double r15897622 = x;
        double r15897623 = y;
        double r15897624 = r15897622 / r15897623;
        double r15897625 = z;
        double r15897626 = r15897625 / r15897623;
        double r15897627 = r15897624 - r15897626;
        double r15897628 = 0.75;
        double r15897629 = r15897627 + r15897628;
        double r15897630 = 4.0;
        double r15897631 = 1.0;
        double r15897632 = fma(r15897629, r15897630, r15897631);
        return r15897632;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - z}{y} + 0.75, 4, 1\right)}\]
  3. Using strategy rm

  4. Applied div-sub0.0

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{x}{y} - \frac{z}{y}\right)} + 0.75, 4, 1\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\left(\frac{x}{y} - \frac{z}{y}\right) + 0.75, 4, 1\right)\]

Reproduce

herbie shell --seed 2019200 +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)))