Average Error: 0.1 → 0.0
Time: 11.3s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x - z}{y} + 0.25, 4.0, 1.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x - z}{y} + 0.25, 4.0, 1.0\right)
double f(double x, double y, double z) {
        double r9958303 = 1.0;
        double r9958304 = 4.0;
        double r9958305 = x;
        double r9958306 = y;
        double r9958307 = 0.25;
        double r9958308 = r9958306 * r9958307;
        double r9958309 = r9958305 + r9958308;
        double r9958310 = z;
        double r9958311 = r9958309 - r9958310;
        double r9958312 = r9958304 * r9958311;
        double r9958313 = r9958312 / r9958306;
        double r9958314 = r9958303 + r9958313;
        return r9958314;
}
↓
double f(double x, double y, double z) {
        double r9958315 = x;
        double r9958316 = z;
        double r9958317 = r9958315 - r9958316;
        double r9958318 = y;
        double r9958319 = r9958317 / r9958318;
        double r9958320 = 0.25;
        double r9958321 = r9958319 + r9958320;
        double r9958322 = 4.0;
        double r9958323 = 1.0;
        double r9958324 = fma(r9958321, r9958322, r9958323);
        return r9958324;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{x - z}{y} + 0.25, 4.0, 1.0\right)\]

Reproduce

herbie shell --seed 2019163 +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)))