Average Error: 0.1 → 0.0
Time: 12.3s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)
double f(double x, double y, double z) {
        double r10436481 = 1.0;
        double r10436482 = 4.0;
        double r10436483 = x;
        double r10436484 = y;
        double r10436485 = 0.25;
        double r10436486 = r10436484 * r10436485;
        double r10436487 = r10436483 + r10436486;
        double r10436488 = z;
        double r10436489 = r10436487 - r10436488;
        double r10436490 = r10436482 * r10436489;
        double r10436491 = r10436490 / r10436484;
        double r10436492 = r10436481 + r10436491;
        return r10436492;
}
↓
double f(double x, double y, double z) {
        double r10436493 = 4.0;
        double r10436494 = x;
        double r10436495 = z;
        double r10436496 = r10436494 - r10436495;
        double r10436497 = y;
        double r10436498 = r10436496 / r10436497;
        double r10436499 = 2.0;
        double r10436500 = fma(r10436493, r10436498, r10436499);
        return r10436500;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4.0, 1.0\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4.0 \cdot \frac{x}{y} + 2.0\right) - 4.0 \cdot \frac{z}{y}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4.0, \frac{x - z}{y}, 2.0\right)\]

Reproduce

herbie shell --seed 2019164 +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)))