Average Error: 0.1 → 0.1
Time: 5.8s
Precision: 64
\[\frac{841}{108} \cdot x + \frac{4}{29}\]
↓
\[\mathsf{fma}\left(x, 7.787037037037037201514522166689857840538, 0.1379310344827586187754775437497301027179\right)\]
\frac{841}{108} \cdot x + \frac{4}{29}
↓
\mathsf{fma}\left(x, 7.787037037037037201514522166689857840538, 0.1379310344827586187754775437497301027179\right)
double f(double x) {
        double r10811669 = 841.0;
        double r10811670 = 108.0;
        double r10811671 = r10811669 / r10811670;
        double r10811672 = x;
        double r10811673 = r10811671 * r10811672;
        double r10811674 = 4.0;
        double r10811675 = 29.0;
        double r10811676 = r10811674 / r10811675;
        double r10811677 = r10811673 + r10811676;
        return r10811677;
}
↓
double f(double x) {
        double r10811678 = x;
        double r10811679 = 7.787037037037037;
        double r10811680 = 0.13793103448275862;
        double r10811681 = fma(r10811678, r10811679, r10811680);
        return r10811681;
}

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

    \[\leadsto \color{blue}{7.787037037037037201514522166689857840538 \cdot x + 0.1379310344827586187754775437497301027179}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 7.787037037037037201514522166689857840538, 0.1379310344827586187754775437497301027179\right)}\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 7.787037037037037201514522166689857840538, 0.1379310344827586187754775437497301027179\right)\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, A"
  (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))