Average Error: 0.2 → 0.0
Time: 12.6s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\left(\frac{x}{y} + 0.75\right) - \frac{z}{y}, 4.0, 1.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(\left(\frac{x}{y} + 0.75\right) - \frac{z}{y}, 4.0, 1.0\right)
double f(double x, double y, double z) {
        double r13607563 = 1.0;
        double r13607564 = 4.0;
        double r13607565 = x;
        double r13607566 = y;
        double r13607567 = 0.75;
        double r13607568 = r13607566 * r13607567;
        double r13607569 = r13607565 + r13607568;
        double r13607570 = z;
        double r13607571 = r13607569 - r13607570;
        double r13607572 = r13607564 * r13607571;
        double r13607573 = r13607572 / r13607566;
        double r13607574 = r13607563 + r13607573;
        return r13607574;
}
↓
double f(double x, double y, double z) {
        double r13607575 = x;
        double r13607576 = y;
        double r13607577 = r13607575 / r13607576;
        double r13607578 = 0.75;
        double r13607579 = r13607577 + r13607578;
        double r13607580 = z;
        double r13607581 = r13607580 / r13607576;
        double r13607582 = r13607579 - r13607581;
        double r13607583 = 4.0;
        double r13607584 = 1.0;
        double r13607585 = fma(r13607582, r13607583, r13607584);
        return r13607585;
}

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

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

  4. Applied div-sub0.0

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

  5. Applied associate-+r-0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +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)))