Average Error: 0.1 → 0.1
Time: 1.7s
Precision: 64
\[\frac{841}{108} \cdot x + \frac{4}{29}\]
↓
\[\mathsf{fma}\left(7.7870370370370372, x, 0.13793103448275862\right)\]
\frac{841}{108} \cdot x + \frac{4}{29}
↓
\mathsf{fma}\left(7.7870370370370372, x, 0.13793103448275862\right)
double f(double x) {
        double r230025 = 841.0;
        double r230026 = 108.0;
        double r230027 = r230025 / r230026;
        double r230028 = x;
        double r230029 = r230027 * r230028;
        double r230030 = 4.0;
        double r230031 = 29.0;
        double r230032 = r230030 / r230031;
        double r230033 = r230029 + r230032;
        return r230033;
}
↓
double f(double x) {
        double r230034 = 7.787037037037037;
        double r230035 = x;
        double r230036 = 0.13793103448275862;
        double r230037 = fma(r230034, r230035, r230036);
        return r230037;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[\frac{841}{108} \cdot x + \frac{4}{29}\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{7.7870370370370372 \cdot x + 0.13793103448275862}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(7.7870370370370372, x, 0.13793103448275862\right)}\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(7.7870370370370372, x, 0.13793103448275862\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, A"
  :precision binary64
  (+ (* (/ 841 108) x) (/ 4 29)))