Average Error: 0.1 → 0.0
Time: 11.4s
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 r270294 = 1.0;
        double r270295 = 4.0;
        double r270296 = x;
        double r270297 = y;
        double r270298 = 0.25;
        double r270299 = r270297 * r270298;
        double r270300 = r270296 + r270299;
        double r270301 = z;
        double r270302 = r270300 - r270301;
        double r270303 = r270295 * r270302;
        double r270304 = r270303 / r270297;
        double r270305 = r270294 + r270304;
        return r270305;
}
↓
double f(double x, double y, double z) {
        double r270306 = 4.0;
        double r270307 = x;
        double r270308 = y;
        double r270309 = r270307 / r270308;
        double r270310 = z;
        double r270311 = r270310 / r270308;
        double r270312 = r270309 - r270311;
        double r270313 = 2.0;
        double r270314 = fma(r270306, r270312, r270313);
        return r270314;
}

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

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

  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 2020045 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))