Average Error: 0.0 → 0.0
Time: 37.0s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right), \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\right)}^{3}}, -x\right) \cdot 0.7071100000000000163069557856942992657423\]
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(\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right), \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\right)}^{3}}, -x\right) \cdot 0.7071100000000000163069557856942992657423
double f(double x) {
        double r89442 = 0.70711;
        double r89443 = 2.30753;
        double r89444 = x;
        double r89445 = 0.27061;
        double r89446 = r89444 * r89445;
        double r89447 = r89443 + r89446;
        double r89448 = 1.0;
        double r89449 = 0.99229;
        double r89450 = 0.04481;
        double r89451 = r89444 * r89450;
        double r89452 = r89449 + r89451;
        double r89453 = r89444 * r89452;
        double r89454 = r89448 + r89453;
        double r89455 = r89447 / r89454;
        double r89456 = r89455 - r89444;
        double r89457 = r89442 * r89456;
        return r89457;
}
↓
double f(double x) {
        double r89458 = 0.27061;
        double r89459 = x;
        double r89460 = 2.30753;
        double r89461 = fma(r89458, r89459, r89460);
        double r89462 = 1.0;
        double r89463 = 0.04481;
        double r89464 = 0.99229;
        double r89465 = fma(r89463, r89459, r89464);
        double r89466 = 1.0;
        double r89467 = fma(r89465, r89459, r89466);
        double r89468 = r89462 / r89467;
        double r89469 = 3.0;
        double r89470 = pow(r89468, r89469);
        double r89471 = cbrt(r89470);
        double r89472 = -r89459;
        double r89473 = fma(r89461, r89471, r89472);
        double r89474 = 0.70711;
        double r89475 = r89473 * r89474;
        return r89475;
}

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

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

  4. Applied div-inv0.0

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

  5. Applied fma-neg0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied cbrt-undiv0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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)))