Average Error: 0.1 → 0.0
Time: 9.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r338330 = 1.0;
        double r338331 = 4.0;
        double r338332 = x;
        double r338333 = y;
        double r338334 = 0.25;
        double r338335 = r338333 * r338334;
        double r338336 = r338332 + r338335;
        double r338337 = z;
        double r338338 = r338336 - r338337;
        double r338339 = r338331 * r338338;
        double r338340 = r338339 / r338333;
        double r338341 = r338330 + r338340;
        return r338341;
}
↓
double f(double x, double y, double z) {
        double r338342 = x;
        double r338343 = y;
        double r338344 = r338342 / r338343;
        double r338345 = z;
        double r338346 = r338345 / r338343;
        double r338347 = r338344 - r338346;
        double r338348 = 4.0;
        double r338349 = 2.0;
        double r338350 = fma(r338347, r338348, r338349);
        return r338350;
}

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

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))