Average Error: 0.2 → 0.0
Time: 3.5s
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 r246196 = 1.0;
        double r246197 = 4.0;
        double r246198 = x;
        double r246199 = y;
        double r246200 = 0.75;
        double r246201 = r246199 * r246200;
        double r246202 = r246198 + r246201;
        double r246203 = z;
        double r246204 = r246202 - r246203;
        double r246205 = r246197 * r246204;
        double r246206 = r246205 / r246199;
        double r246207 = r246196 + r246206;
        return r246207;
}
↓
double f(double x, double y, double z) {
        double r246208 = 4.0;
        double r246209 = x;
        double r246210 = y;
        double r246211 = r246209 / r246210;
        double r246212 = z;
        double r246213 = r246212 / r246210;
        double r246214 = r246208 * r246213;
        double r246215 = r246208 - r246214;
        double r246216 = fma(r246208, r246211, r246215);
        return r246216;
}

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 2020025 +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)))