Average Error: 0.1 → 0.0
Time: 11.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r17767534 = 1.0;
        double r17767535 = 4.0;
        double r17767536 = x;
        double r17767537 = y;
        double r17767538 = 0.25;
        double r17767539 = r17767537 * r17767538;
        double r17767540 = r17767536 + r17767539;
        double r17767541 = z;
        double r17767542 = r17767540 - r17767541;
        double r17767543 = r17767535 * r17767542;
        double r17767544 = r17767543 / r17767537;
        double r17767545 = r17767534 + r17767544;
        return r17767545;
}
↓
double f(double x, double y, double z) {
        double r17767546 = 4.0;
        double r17767547 = x;
        double r17767548 = y;
        double r17767549 = r17767547 / r17767548;
        double r17767550 = z;
        double r17767551 = r17767550 / r17767548;
        double r17767552 = r17767549 - r17767551;
        double r17767553 = 2.0;
        double r17767554 = fma(r17767546, r17767552, r17767553);
        return r17767554;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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)))