Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)
double f(double x) {
        double r108478 = 0.70711;
        double r108479 = 2.30753;
        double r108480 = x;
        double r108481 = 0.27061;
        double r108482 = r108480 * r108481;
        double r108483 = r108479 + r108482;
        double r108484 = 1.0;
        double r108485 = 0.99229;
        double r108486 = 0.04481;
        double r108487 = r108480 * r108486;
        double r108488 = r108485 + r108487;
        double r108489 = r108480 * r108488;
        double r108490 = r108484 + r108489;
        double r108491 = r108483 / r108490;
        double r108492 = r108491 - r108480;
        double r108493 = r108478 * r108492;
        return r108493;
}
↓
double f(double x) {
        double r108494 = x;
        double r108495 = -r108494;
        double r108496 = 0.70711;
        double r108497 = 0.27061;
        double r108498 = 2.30753;
        double r108499 = fma(r108497, r108494, r108498);
        double r108500 = r108496 * r108499;
        double r108501 = 0.04481;
        double r108502 = 0.99229;
        double r108503 = fma(r108501, r108494, r108502);
        double r108504 = 1.0;
        double r108505 = fma(r108494, r108503, r108504);
        double r108506 = r108500 / r108505;
        double r108507 = fma(r108495, r108496, r108506);
        return r108507;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))