Average Error: 0.1 → 0.0
Time: 6.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, 0.25 - \frac{z - x}{y}, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, 0.25 - \frac{z - x}{y}, 1\right)
double f(double x, double y, double z) {
        double r160062 = 1.0;
        double r160063 = 4.0;
        double r160064 = x;
        double r160065 = y;
        double r160066 = 0.25;
        double r160067 = r160065 * r160066;
        double r160068 = r160064 + r160067;
        double r160069 = z;
        double r160070 = r160068 - r160069;
        double r160071 = r160063 * r160070;
        double r160072 = r160071 / r160065;
        double r160073 = r160062 + r160072;
        return r160073;
}
↓
double f(double x, double y, double z) {
        double r160074 = 4.0;
        double r160075 = 0.25;
        double r160076 = z;
        double r160077 = x;
        double r160078 = r160076 - r160077;
        double r160079 = y;
        double r160080 = r160078 / r160079;
        double r160081 = r160075 - r160080;
        double r160082 = 1.0;
        double r160083 = fma(r160074, r160081, r160082);
        return r160083;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019235 +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)))