Average Error: 0.0 → 0.0
Time: 2.9s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\left(\frac{1 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)\right)}^{3}}} - x\right) + \mathsf{fma}\left(-x, 1, x\right)\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\left(\frac{1 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)\right)}^{3}}} - x\right) + \mathsf{fma}\left(-x, 1, x\right)
double f(double x) {
        double r87851 = 2.30753;
        double r87852 = x;
        double r87853 = 0.27061;
        double r87854 = r87852 * r87853;
        double r87855 = r87851 + r87854;
        double r87856 = 1.0;
        double r87857 = 0.99229;
        double r87858 = 0.04481;
        double r87859 = r87852 * r87858;
        double r87860 = r87857 + r87859;
        double r87861 = r87852 * r87860;
        double r87862 = r87856 + r87861;
        double r87863 = r87855 / r87862;
        double r87864 = r87863 - r87852;
        return r87864;
}
↓
double f(double x) {
        double r87865 = 1.0;
        double r87866 = 0.27061;
        double r87867 = x;
        double r87868 = 2.30753;
        double r87869 = fma(r87866, r87867, r87868);
        double r87870 = r87865 * r87869;
        double r87871 = 0.04481;
        double r87872 = 0.99229;
        double r87873 = fma(r87871, r87867, r87872);
        double r87874 = 1.0;
        double r87875 = fma(r87867, r87873, r87874);
        double r87876 = 3.0;
        double r87877 = pow(r87875, r87876);
        double r87878 = cbrt(r87877);
        double r87879 = r87870 / r87878;
        double r87880 = r87879 - r87867;
        double r87881 = -r87867;
        double r87882 = fma(r87881, r87865, r87867);
        double r87883 = r87880 + r87882;
        return r87883;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.7

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

  4. Applied add-sqr-sqrt8.6

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

  5. Applied prod-diff8.6

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  9. Applied add-cbrt-cube0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x))