Average Error: 0.2 → 0.0
Time: 16.6s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)
double f(double x, double y, double z) {
        double r229258 = 1.0;
        double r229259 = 4.0;
        double r229260 = x;
        double r229261 = y;
        double r229262 = 0.75;
        double r229263 = r229261 * r229262;
        double r229264 = r229260 + r229263;
        double r229265 = z;
        double r229266 = r229264 - r229265;
        double r229267 = r229259 * r229266;
        double r229268 = r229267 / r229261;
        double r229269 = r229258 + r229268;
        return r229269;
}
↓
double f(double x, double y, double z) {
        double r229270 = 0.75;
        double r229271 = x;
        double r229272 = y;
        double r229273 = r229271 / r229272;
        double r229274 = z;
        double r229275 = r229274 / r229272;
        double r229276 = r229273 - r229275;
        double r229277 = r229270 + r229276;
        double r229278 = 4.0;
        double r229279 = 1.0;
        double r229280 = fma(r229277, r229278, r229279);
        return r229280;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 + \frac{x - z}{y}, 4, 1\right)}\]
  3. Using strategy rm

  4. Applied div-sub0.0

    \[\leadsto \mathsf{fma}\left(0.75 + \color{blue}{\left(\frac{x}{y} - \frac{z}{y}\right)}, 4, 1\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)\]

Reproduce

herbie shell --seed 2019325 +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)))