Average Error: 0.2 → 0.0
Time: 13.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)
double f(double x, double y, double z) {
        double r184538 = 1.0;
        double r184539 = 4.0;
        double r184540 = x;
        double r184541 = y;
        double r184542 = 0.75;
        double r184543 = r184541 * r184542;
        double r184544 = r184540 + r184543;
        double r184545 = z;
        double r184546 = r184544 - r184545;
        double r184547 = r184539 * r184546;
        double r184548 = r184547 / r184541;
        double r184549 = r184538 + r184548;
        return r184549;
}
↓
double f(double x, double y, double z) {
        double r184550 = 4.0;
        double r184551 = x;
        double r184552 = y;
        double r184553 = r184551 / r184552;
        double r184554 = z;
        double r184555 = r184554 / r184552;
        double r184556 = r184553 - r184555;
        double r184557 = fma(r184550, r184556, r184550);
        return r184557;
}

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

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

  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

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))