Average Error: 0.1 → 0.0
Time: 1.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r232044 = 1.0;
        double r232045 = 4.0;
        double r232046 = x;
        double r232047 = y;
        double r232048 = 0.75;
        double r232049 = r232047 * r232048;
        double r232050 = r232046 + r232049;
        double r232051 = z;
        double r232052 = r232050 - r232051;
        double r232053 = r232045 * r232052;
        double r232054 = r232053 / r232047;
        double r232055 = r232044 + r232054;
        return r232055;
}
↓
double f(double x, double y, double z) {
        double r232056 = 4.0;
        double r232057 = x;
        double r232058 = y;
        double r232059 = r232057 / r232058;
        double r232060 = z;
        double r232061 = r232060 / r232058;
        double r232062 = r232056 * r232061;
        double r232063 = r232056 - r232062;
        double r232064 = fma(r232056, r232059, r232063);
        return r232064;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))