Average Error: 0.0 → 0.0
Time: 1.9s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}\right)\]
0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}\right)
double f(double x) {
        double r119393 = 0.70711;
        double r119394 = 2.30753;
        double r119395 = x;
        double r119396 = 0.27061;
        double r119397 = r119395 * r119396;
        double r119398 = r119394 + r119397;
        double r119399 = 1.0;
        double r119400 = 0.99229;
        double r119401 = 0.04481;
        double r119402 = r119395 * r119401;
        double r119403 = r119400 + r119402;
        double r119404 = r119395 * r119403;
        double r119405 = r119399 + r119404;
        double r119406 = r119398 / r119405;
        double r119407 = r119406 - r119395;
        double r119408 = r119393 * r119407;
        return r119408;
}
↓
double f(double x) {
        double r119409 = x;
        double r119410 = -r119409;
        double r119411 = 0.70711;
        double r119412 = 1.0;
        double r119413 = 0.04481;
        double r119414 = 0.99229;
        double r119415 = fma(r119413, r119409, r119414);
        double r119416 = 1.0;
        double r119417 = fma(r119409, r119415, r119416);
        double r119418 = 0.27061;
        double r119419 = 2.30753;
        double r119420 = fma(r119418, r119409, r119419);
        double r119421 = r119411 * r119420;
        double r119422 = r119417 / r119421;
        double r119423 = r119412 / r119422;
        double r119424 = fma(r119410, r119411, r119423);
        return r119424;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}\]
  3. Using strategy rm

  4. Applied clear-num0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}\right)\]

Reproduce

herbie shell --seed 2019354 +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)))