Average Error: 0.1 → 0.0
Time: 12.9s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[4.0 \cdot \left(\frac{x}{y} - \frac{z}{y}\right) + 2.0\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
4.0 \cdot \left(\frac{x}{y} - \frac{z}{y}\right) + 2.0
double f(double x, double y, double z) {
        double r4551223 = 1.0;
        double r4551224 = 4.0;
        double r4551225 = x;
        double r4551226 = y;
        double r4551227 = 0.25;
        double r4551228 = r4551226 * r4551227;
        double r4551229 = r4551225 + r4551228;
        double r4551230 = z;
        double r4551231 = r4551229 - r4551230;
        double r4551232 = r4551224 * r4551231;
        double r4551233 = r4551232 / r4551226;
        double r4551234 = r4551223 + r4551233;
        return r4551234;
}
↓
double f(double x, double y, double z) {
        double r4551235 = 4.0;
        double r4551236 = x;
        double r4551237 = y;
        double r4551238 = r4551236 / r4551237;
        double r4551239 = z;
        double r4551240 = r4551239 / r4551237;
        double r4551241 = r4551238 - r4551240;
        double r4551242 = r4551235 * r4551241;
        double r4551243 = 2.0;
        double r4551244 = r4551242 + r4551243;
        return r4551244;
}

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

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019156 +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)))