Average Error: 0.2 → 0.0
Time: 14.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)
double f(double x, double y, double z) {
        double r176084 = 1.0;
        double r176085 = 4.0;
        double r176086 = x;
        double r176087 = y;
        double r176088 = 0.75;
        double r176089 = r176087 * r176088;
        double r176090 = r176086 + r176089;
        double r176091 = z;
        double r176092 = r176090 - r176091;
        double r176093 = r176085 * r176092;
        double r176094 = r176093 / r176087;
        double r176095 = r176084 + r176094;
        return r176095;
}
↓
double f(double x, double y, double z) {
        double r176096 = 4.0;
        double r176097 = x;
        double r176098 = z;
        double r176099 = r176097 - r176098;
        double r176100 = y;
        double r176101 = r176099 / r176100;
        double r176102 = fma(r176096, r176101, r176096);
        return r176102;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +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)))