Average Error: 0.1 → 0.0
Time: 17.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, 4, 2 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y}, 4, 2 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r13380039 = 1.0;
        double r13380040 = 4.0;
        double r13380041 = x;
        double r13380042 = y;
        double r13380043 = 0.25;
        double r13380044 = r13380042 * r13380043;
        double r13380045 = r13380041 + r13380044;
        double r13380046 = z;
        double r13380047 = r13380045 - r13380046;
        double r13380048 = r13380040 * r13380047;
        double r13380049 = r13380048 / r13380042;
        double r13380050 = r13380039 + r13380049;
        return r13380050;
}
↓
double f(double x, double y, double z) {
        double r13380051 = x;
        double r13380052 = y;
        double r13380053 = r13380051 / r13380052;
        double r13380054 = 4.0;
        double r13380055 = 2.0;
        double r13380056 = z;
        double r13380057 = r13380056 / r13380052;
        double r13380058 = r13380054 * r13380057;
        double r13380059 = r13380055 - r13380058;
        double r13380060 = fma(r13380053, r13380054, r13380059);
        return r13380060;
}

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. Using strategy rm

  4. Applied div-sub0.0

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

  5. Applied associate--r-0.0

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

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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)))