Average Error: 0.0 → 0.0
Time: 39.5s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812, \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}^{3}}, -x\right)\]
0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)
↓
0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812, \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}^{3}}, -x\right)
double f(double x) {
        double r68260 = 0.70711;
        double r68261 = 2.30753;
        double r68262 = x;
        double r68263 = 0.27061;
        double r68264 = r68262 * r68263;
        double r68265 = r68261 + r68264;
        double r68266 = 1.0;
        double r68267 = 0.99229;
        double r68268 = 0.04481;
        double r68269 = r68262 * r68268;
        double r68270 = r68267 + r68269;
        double r68271 = r68262 * r68270;
        double r68272 = r68266 + r68271;
        double r68273 = r68265 / r68272;
        double r68274 = r68273 - r68262;
        double r68275 = r68260 * r68274;
        return r68275;
}
↓
double f(double x) {
        double r68276 = 0.70711;
        double r68277 = 2.30753;
        double r68278 = x;
        double r68279 = 0.27061;
        double r68280 = r68278 * r68279;
        double r68281 = r68277 + r68280;
        double r68282 = 1.0;
        double r68283 = 0.04481;
        double r68284 = 0.99229;
        double r68285 = fma(r68278, r68283, r68284);
        double r68286 = 1.0;
        double r68287 = fma(r68278, r68285, r68286);
        double r68288 = r68282 / r68287;
        double r68289 = 3.0;
        double r68290 = pow(r68288, r68289);
        double r68291 = cbrt(r68290);
        double r68292 = -r68278;
        double r68293 = fma(r68281, r68291, r68292);
        double r68294 = r68276 * r68293;
        return r68294;
}

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. Using strategy rm

  3. Applied div-inv0.0

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

  4. Applied fma-neg0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Applied cbrt-undiv0.0

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

  9. Simplified0.0

    \[\leadsto 0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812, \sqrt[3]{\color{blue}{{\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}^{3}}}, -x\right)\]

  10. Final simplification0.0

    \[\leadsto 0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812, \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}^{3}}, -x\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.707110000000000016 (- (/ (+ 2.30753 (* x 0.27061000000000002)) (+ 1 (* x (+ 0.992290000000000005 (* x 0.044810000000000003))))) x)))