Average Error: 0.1 → 0.0
Time: 1.4s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y}, 1 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y}, 1 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r239595 = 1.0;
        double r239596 = 4.0;
        double r239597 = x;
        double r239598 = y;
        double r239599 = 0.25;
        double r239600 = r239598 * r239599;
        double r239601 = r239597 + r239600;
        double r239602 = z;
        double r239603 = r239601 - r239602;
        double r239604 = r239596 * r239603;
        double r239605 = r239604 / r239598;
        double r239606 = r239595 + r239605;
        return r239606;
}
↓
double f(double x, double y, double z) {
        double r239607 = 1.0;
        double r239608 = 4.0;
        double r239609 = x;
        double r239610 = y;
        double r239611 = r239609 / r239610;
        double r239612 = z;
        double r239613 = r239612 / r239610;
        double r239614 = r239608 * r239613;
        double r239615 = r239607 - r239614;
        double r239616 = fma(r239608, r239611, r239615);
        double r239617 = r239607 + r239616;
        return r239617;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))