Average Error: 0.2 → 0.0
Time: 1.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r198491 = 1.0;
        double r198492 = 4.0;
        double r198493 = x;
        double r198494 = y;
        double r198495 = 0.75;
        double r198496 = r198494 * r198495;
        double r198497 = r198493 + r198496;
        double r198498 = z;
        double r198499 = r198497 - r198498;
        double r198500 = r198492 * r198499;
        double r198501 = r198500 / r198494;
        double r198502 = r198491 + r198501;
        return r198502;
}
↓
double f(double x, double y, double z) {
        double r198503 = 4.0;
        double r198504 = x;
        double r198505 = y;
        double r198506 = r198504 / r198505;
        double r198507 = z;
        double r198508 = r198507 / r198505;
        double r198509 = r198503 * r198508;
        double r198510 = r198503 - r198509;
        double r198511 = fma(r198503, r198506, r198510);
        return r198511;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))