Average Error: 0.0 → 0.0
Time: 34.5s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\left(\left(\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right) \cdot \left(\sqrt[3]{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}} \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}\right)\right) \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}} - x\right) \cdot 0.7071100000000000163069557856942992657423\]
0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)
↓
\left(\left(\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right) \cdot \left(\sqrt[3]{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}} \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}\right)\right) \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}} - x\right) \cdot 0.7071100000000000163069557856942992657423
double f(double x) {
        double r74928 = 0.70711;
        double r74929 = 2.30753;
        double r74930 = x;
        double r74931 = 0.27061;
        double r74932 = r74930 * r74931;
        double r74933 = r74929 + r74932;
        double r74934 = 1.0;
        double r74935 = 0.99229;
        double r74936 = 0.04481;
        double r74937 = r74930 * r74936;
        double r74938 = r74935 + r74937;
        double r74939 = r74930 * r74938;
        double r74940 = r74934 + r74939;
        double r74941 = r74933 / r74940;
        double r74942 = r74941 - r74930;
        double r74943 = r74928 * r74942;
        return r74943;
}
↓
double f(double x) {
        double r74944 = 0.27061;
        double r74945 = x;
        double r74946 = 2.30753;
        double r74947 = fma(r74944, r74945, r74946);
        double r74948 = 1.0;
        double r74949 = 0.04481;
        double r74950 = 0.99229;
        double r74951 = fma(r74949, r74945, r74950);
        double r74952 = 1.0;
        double r74953 = fma(r74951, r74945, r74952);
        double r74954 = r74948 / r74953;
        double r74955 = cbrt(r74954);
        double r74956 = r74955 * r74955;
        double r74957 = r74947 * r74956;
        double r74958 = r74957 * r74955;
        double r74959 = r74958 - r74945;
        double r74960 = 0.70711;
        double r74961 = r74959 * r74960;
        return r74961;
}

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. Using strategy rm

  6. Applied add-cube-cbrt0.0

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

  7. Applied associate-*r*0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 +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)))