Average Error: 0.2 → 0.0
Time: 2.8s
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 r245986 = 1.0;
        double r245987 = 4.0;
        double r245988 = x;
        double r245989 = y;
        double r245990 = 0.75;
        double r245991 = r245989 * r245990;
        double r245992 = r245988 + r245991;
        double r245993 = z;
        double r245994 = r245992 - r245993;
        double r245995 = r245987 * r245994;
        double r245996 = r245995 / r245989;
        double r245997 = r245986 + r245996;
        return r245997;
}
↓
double f(double x, double y, double z) {
        double r245998 = 4.0;
        double r245999 = x;
        double r246000 = y;
        double r246001 = r245999 / r246000;
        double r246002 = z;
        double r246003 = r246002 / r246000;
        double r246004 = r245998 * r246003;
        double r246005 = r245998 - r246004;
        double r246006 = fma(r245998, r246001, r246005);
        return r246006;
}

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 2020089 +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)))