Average Error: 0.0 → 0.0
Time: 39.0s
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 r67257 = 0.70711;
        double r67258 = 2.30753;
        double r67259 = x;
        double r67260 = 0.27061;
        double r67261 = r67259 * r67260;
        double r67262 = r67258 + r67261;
        double r67263 = 1.0;
        double r67264 = 0.99229;
        double r67265 = 0.04481;
        double r67266 = r67259 * r67265;
        double r67267 = r67264 + r67266;
        double r67268 = r67259 * r67267;
        double r67269 = r67263 + r67268;
        double r67270 = r67262 / r67269;
        double r67271 = r67270 - r67259;
        double r67272 = r67257 * r67271;
        return r67272;
}
↓
double f(double x) {
        double r67273 = 0.70711;
        double r67274 = 2.30753;
        double r67275 = x;
        double r67276 = 0.27061;
        double r67277 = r67275 * r67276;
        double r67278 = r67274 + r67277;
        double r67279 = 1.0;
        double r67280 = 0.04481;
        double r67281 = 0.99229;
        double r67282 = fma(r67275, r67280, r67281);
        double r67283 = 1.0;
        double r67284 = fma(r67275, r67282, r67283);
        double r67285 = r67279 / r67284;
        double r67286 = 3.0;
        double r67287 = pow(r67285, r67286);
        double r67288 = cbrt(r67287);
        double r67289 = -r67275;
        double r67290 = fma(r67278, r67288, r67289);
        double r67291 = r67273 * r67290;
        return r67291;
}

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)))