Average Error: 0.0 → 0.0
Time: 15.4s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{0.2706100000000000171951342053944244980812 \cdot x + 2.307529999999999859028321225196123123169}{1 + \left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right) \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} - x\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\frac{0.2706100000000000171951342053944244980812 \cdot x + 2.307529999999999859028321225196123123169}{1 + \left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right) \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} - x
double f(double x) {
        double r1928115 = 2.30753;
        double r1928116 = x;
        double r1928117 = 0.27061;
        double r1928118 = r1928116 * r1928117;
        double r1928119 = r1928115 + r1928118;
        double r1928120 = 1.0;
        double r1928121 = 0.99229;
        double r1928122 = 0.04481;
        double r1928123 = r1928116 * r1928122;
        double r1928124 = r1928121 + r1928123;
        double r1928125 = r1928116 * r1928124;
        double r1928126 = r1928120 + r1928125;
        double r1928127 = r1928119 / r1928126;
        double r1928128 = r1928127 - r1928116;
        return r1928128;
}
↓
double f(double x) {
        double r1928129 = 0.27061;
        double r1928130 = x;
        double r1928131 = r1928129 * r1928130;
        double r1928132 = 2.30753;
        double r1928133 = r1928131 + r1928132;
        double r1928134 = 1.0;
        double r1928135 = 0.04481;
        double r1928136 = 0.99229;
        double r1928137 = fma(r1928130, r1928135, r1928136);
        double r1928138 = cbrt(r1928130);
        double r1928139 = r1928137 * r1928138;
        double r1928140 = r1928138 * r1928138;
        double r1928141 = r1928139 * r1928140;
        double r1928142 = r1928134 + r1928141;
        double r1928143 = r1928133 / r1928142;
        double r1928144 = r1928143 - r1928130;
        return r1928144;
}

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-cbrt0.0

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

  4. Applied associate-*l*0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))