Average Error: 0.2 → 0.0
Time: 15.4s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)
double f(double x, double y, double z) {
        double r24676 = 1.0;
        double r24677 = 4.0;
        double r24678 = x;
        double r24679 = y;
        double r24680 = 0.75;
        double r24681 = r24679 * r24680;
        double r24682 = r24678 + r24681;
        double r24683 = z;
        double r24684 = r24682 - r24683;
        double r24685 = r24677 * r24684;
        double r24686 = r24685 / r24679;
        double r24687 = r24676 + r24686;
        return r24687;
}
↓
double f(double x, double y, double z) {
        double r24688 = 0.75;
        double r24689 = x;
        double r24690 = y;
        double r24691 = r24689 / r24690;
        double r24692 = z;
        double r24693 = r24692 / r24690;
        double r24694 = r24691 - r24693;
        double r24695 = r24688 + r24694;
        double r24696 = 4.0;
        double r24697 = 1.0;
        double r24698 = fma(r24695, r24696, r24697);
        return r24698;
}

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

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

  4. Applied div-sub0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019310 +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)))