Average Error: 0.1 → 0.0
Time: 20.9s
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 r34131236 = 1.0;
        double r34131237 = 4.0;
        double r34131238 = x;
        double r34131239 = y;
        double r34131240 = 0.25;
        double r34131241 = r34131239 * r34131240;
        double r34131242 = r34131238 + r34131241;
        double r34131243 = z;
        double r34131244 = r34131242 - r34131243;
        double r34131245 = r34131237 * r34131244;
        double r34131246 = r34131245 / r34131239;
        double r34131247 = r34131236 + r34131246;
        return r34131247;
}
↓
double f(double x, double y, double z) {
        double r34131248 = x;
        double r34131249 = y;
        double r34131250 = r34131248 / r34131249;
        double r34131251 = z;
        double r34131252 = r34131251 / r34131249;
        double r34131253 = r34131250 - r34131252;
        double r34131254 = 4.0;
        double r34131255 = 2.0;
        double r34131256 = fma(r34131253, r34131254, r34131255);
        return r34131256;
}

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

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

  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 2019173 +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)))