Average Error: 0.1 → 0.0
Time: 9.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, 4, 2 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y}, 4, 2 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r181507 = 1.0;
        double r181508 = 4.0;
        double r181509 = x;
        double r181510 = y;
        double r181511 = 0.25;
        double r181512 = r181510 * r181511;
        double r181513 = r181509 + r181512;
        double r181514 = z;
        double r181515 = r181513 - r181514;
        double r181516 = r181508 * r181515;
        double r181517 = r181516 / r181510;
        double r181518 = r181507 + r181517;
        return r181518;
}
↓
double f(double x, double y, double z) {
        double r181519 = x;
        double r181520 = y;
        double r181521 = r181519 / r181520;
        double r181522 = 4.0;
        double r181523 = 2.0;
        double r181524 = z;
        double r181525 = r181524 / r181520;
        double r181526 = r181522 * r181525;
        double r181527 = r181523 - r181526;
        double r181528 = fma(r181521, r181522, r181527);
        return r181528;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, 4, 2 - \frac{z}{y} \cdot 4\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, 4, 2 - 4 \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))