Average Error: 0.2 → 0.0
Time: 5.0s
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 r233011 = 1.0;
        double r233012 = 4.0;
        double r233013 = x;
        double r233014 = y;
        double r233015 = 0.75;
        double r233016 = r233014 * r233015;
        double r233017 = r233013 + r233016;
        double r233018 = z;
        double r233019 = r233017 - r233018;
        double r233020 = r233012 * r233019;
        double r233021 = r233020 / r233014;
        double r233022 = r233011 + r233021;
        return r233022;
}
↓
double f(double x, double y, double z) {
        double r233023 = 4.0;
        double r233024 = x;
        double r233025 = y;
        double r233026 = r233024 / r233025;
        double r233027 = z;
        double r233028 = r233027 / r233025;
        double r233029 = r233023 * r233028;
        double r233030 = r233023 - r233029;
        double r233031 = fma(r233023, r233026, r233030);
        return r233031;
}

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 2020036 +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)))