Average Error: 0.2 → 0.0
Time: 1.8s
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 r234067 = 1.0;
        double r234068 = 4.0;
        double r234069 = x;
        double r234070 = y;
        double r234071 = 0.75;
        double r234072 = r234070 * r234071;
        double r234073 = r234069 + r234072;
        double r234074 = z;
        double r234075 = r234073 - r234074;
        double r234076 = r234068 * r234075;
        double r234077 = r234076 / r234070;
        double r234078 = r234067 + r234077;
        return r234078;
}
↓
double f(double x, double y, double z) {
        double r234079 = 4.0;
        double r234080 = x;
        double r234081 = y;
        double r234082 = r234080 / r234081;
        double r234083 = z;
        double r234084 = r234083 / r234081;
        double r234085 = r234079 * r234084;
        double r234086 = r234079 - r234085;
        double r234087 = fma(r234079, r234082, r234086);
        return r234087;
}

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. 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 2020021 +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)))