Average Error: 0.2 → 0.0
Time: 12.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4\right) - 4 \cdot \frac{z}{y}\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4\right) - 4 \cdot \frac{z}{y}
double f(double x, double y, double z) {
        double r168419 = 1.0;
        double r168420 = 4.0;
        double r168421 = x;
        double r168422 = y;
        double r168423 = 0.75;
        double r168424 = r168422 * r168423;
        double r168425 = r168421 + r168424;
        double r168426 = z;
        double r168427 = r168425 - r168426;
        double r168428 = r168420 * r168427;
        double r168429 = r168428 / r168422;
        double r168430 = r168419 + r168429;
        return r168430;
}
↓
double f(double x, double y, double z) {
        double r168431 = 4.0;
        double r168432 = x;
        double r168433 = y;
        double r168434 = r168432 / r168433;
        double r168435 = fma(r168431, r168434, r168431);
        double r168436 = z;
        double r168437 = r168436 / r168433;
        double r168438 = r168431 * r168437;
        double r168439 = r168435 - r168438;
        return r168439;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, 0.75 - \frac{z - x}{y}, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019235 +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)))