Average Error: 0.2 → 0.0
Time: 2.0s
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 r279957 = 1.0;
        double r279958 = 4.0;
        double r279959 = x;
        double r279960 = y;
        double r279961 = 0.75;
        double r279962 = r279960 * r279961;
        double r279963 = r279959 + r279962;
        double r279964 = z;
        double r279965 = r279963 - r279964;
        double r279966 = r279958 * r279965;
        double r279967 = r279966 / r279960;
        double r279968 = r279957 + r279967;
        return r279968;
}
↓
double f(double x, double y, double z) {
        double r279969 = 4.0;
        double r279970 = x;
        double r279971 = y;
        double r279972 = r279970 / r279971;
        double r279973 = z;
        double r279974 = r279973 / r279971;
        double r279975 = r279969 * r279974;
        double r279976 = r279969 - r279975;
        double r279977 = fma(r279969, r279972, r279976);
        return r279977;
}

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 2020046 +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)))