Average Error: 0.1 → 0.0
Time: 12.6s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r16061296 = 1.0;
        double r16061297 = 4.0;
        double r16061298 = x;
        double r16061299 = y;
        double r16061300 = 0.25;
        double r16061301 = r16061299 * r16061300;
        double r16061302 = r16061298 + r16061301;
        double r16061303 = z;
        double r16061304 = r16061302 - r16061303;
        double r16061305 = r16061297 * r16061304;
        double r16061306 = r16061305 / r16061299;
        double r16061307 = r16061296 + r16061306;
        return r16061307;
}
↓
double f(double x, double y, double z) {
        double r16061308 = 4.0;
        double r16061309 = x;
        double r16061310 = y;
        double r16061311 = r16061309 / r16061310;
        double r16061312 = z;
        double r16061313 = r16061312 / r16061310;
        double r16061314 = r16061311 - r16061313;
        double r16061315 = 2.0;
        double r16061316 = fma(r16061308, r16061314, r16061315);
        return r16061316;
}

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

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))