Average Error: 0.1 → 0.0
Time: 13.0s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)
double f(double x, double y, double z) {
        double r15146489 = 1.0;
        double r15146490 = 4.0;
        double r15146491 = x;
        double r15146492 = y;
        double r15146493 = 0.25;
        double r15146494 = r15146492 * r15146493;
        double r15146495 = r15146491 + r15146494;
        double r15146496 = z;
        double r15146497 = r15146495 - r15146496;
        double r15146498 = r15146490 * r15146497;
        double r15146499 = r15146498 / r15146492;
        double r15146500 = r15146489 + r15146499;
        return r15146500;
}
↓
double f(double x, double y, double z) {
        double r15146501 = 4.0;
        double r15146502 = x;
        double r15146503 = z;
        double r15146504 = r15146502 - r15146503;
        double r15146505 = y;
        double r15146506 = r15146504 / r15146505;
        double r15146507 = 2.0;
        double r15146508 = fma(r15146501, r15146506, r15146507);
        return r15146508;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4.0, 1.0\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4.0 \cdot \frac{x}{y} + 2.0\right) - 4.0 \cdot \frac{z}{y}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)\]

Reproduce

herbie shell --seed 2019158 +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)))