Average Error: 0.1 → 0.2
Time: 1.3s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{4}{y}, x - z, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{4}{y}, x - z, 2\right)
double code(double x, double y, double z) {
	return ((double) (1.0 + ((double) (((double) (4.0 * ((double) (((double) (x + ((double) (y * 0.25)))) - z)))) / y))));
}
↓
double code(double x, double y, double z) {
	return ((double) fma(((double) (4.0 / y)), ((double) (x - z)), 2.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.2

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020121 +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)))