Average Error: 0.2 → 0.0
Time: 4.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r261369 = 1.0;
        double r261370 = 4.0;
        double r261371 = x;
        double r261372 = y;
        double r261373 = 0.75;
        double r261374 = r261372 * r261373;
        double r261375 = r261371 + r261374;
        double r261376 = z;
        double r261377 = r261375 - r261376;
        double r261378 = r261370 * r261377;
        double r261379 = r261378 / r261372;
        double r261380 = r261369 + r261379;
        return r261380;
}
↓
double f(double x, double y, double z) {
        double r261381 = 1.0;
        double r261382 = 4.0;
        double r261383 = x;
        double r261384 = y;
        double r261385 = r261383 / r261384;
        double r261386 = 3.0;
        double r261387 = z;
        double r261388 = r261387 / r261384;
        double r261389 = r261382 * r261388;
        double r261390 = r261386 - r261389;
        double r261391 = fma(r261382, r261385, r261390);
        double r261392 = r261381 + r261391;
        return r261392;
}

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. Taylor expanded around 0 0.0

    \[\leadsto 1 + \color{blue}{\left(\left(4 \cdot \frac{x}{y} + 3\right) - 4 \cdot \frac{z}{y}\right)}\]

  3. Simplified0.0

    \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)}\]

  4. Final simplification0.0

    \[\leadsto 1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2020001 +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)))