Average Error: 0.1 → 0.0
Time: 2.3s
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 r251244 = 1.0;
        double r251245 = 4.0;
        double r251246 = x;
        double r251247 = y;
        double r251248 = 0.25;
        double r251249 = r251247 * r251248;
        double r251250 = r251246 + r251249;
        double r251251 = z;
        double r251252 = r251250 - r251251;
        double r251253 = r251245 * r251252;
        double r251254 = r251253 / r251247;
        double r251255 = r251244 + r251254;
        return r251255;
}
↓
double f(double x, double y, double z) {
        double r251256 = 4.0;
        double r251257 = x;
        double r251258 = y;
        double r251259 = r251257 / r251258;
        double r251260 = z;
        double r251261 = r251260 / r251258;
        double r251262 = r251259 - r251261;
        double r251263 = 2.0;
        double r251264 = fma(r251256, r251262, r251263);
        return r251264;
}

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. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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)))