Average Error: 0.1 → 0.0
Time: 9.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)
double f(double x, double y, double z) {
        double r312567 = 1.0;
        double r312568 = 4.0;
        double r312569 = x;
        double r312570 = y;
        double r312571 = 0.25;
        double r312572 = r312570 * r312571;
        double r312573 = r312569 + r312572;
        double r312574 = z;
        double r312575 = r312573 - r312574;
        double r312576 = r312568 * r312575;
        double r312577 = r312576 / r312570;
        double r312578 = r312567 + r312577;
        return r312578;
}
↓
double f(double x, double y, double z) {
        double r312579 = 0.25;
        double r312580 = x;
        double r312581 = z;
        double r312582 = r312580 - r312581;
        double r312583 = y;
        double r312584 = r312582 / r312583;
        double r312585 = r312579 + r312584;
        double r312586 = 4.0;
        double r312587 = 1.0;
        double r312588 = fma(r312585, r312586, r312587);
        return r312588;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)\]

Reproduce

herbie shell --seed 2019351 +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)))