Average Error: 0.0 → 0.0
Time: 12.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 2\right) - 4 \cdot \frac{z}{y}\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 2\right) - 4 \cdot \frac{z}{y}
double f(double x, double y, double z) {
        double r229655 = 1.0;
        double r229656 = 4.0;
        double r229657 = x;
        double r229658 = y;
        double r229659 = 0.25;
        double r229660 = r229658 * r229659;
        double r229661 = r229657 + r229660;
        double r229662 = z;
        double r229663 = r229661 - r229662;
        double r229664 = r229656 * r229663;
        double r229665 = r229664 / r229658;
        double r229666 = r229655 + r229665;
        return r229666;
}
↓
double f(double x, double y, double z) {
        double r229667 = 4.0;
        double r229668 = x;
        double r229669 = y;
        double r229670 = r229668 / r229669;
        double r229671 = 2.0;
        double r229672 = fma(r229667, r229670, r229671);
        double r229673 = z;
        double r229674 = r229673 / r229669;
        double r229675 = r229667 * r229674;
        double r229676 = r229672 - r229675;
        return r229676;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +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)))