Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C

Average Error: 0.0 → 0.0

Time: 17.8s

Precision: 64

Math

\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]

↓

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

C

double f(double x) {
        double r3956126 = 2.30753;
        double r3956127 = x;
        double r3956128 = 0.27061;
        double r3956129 = r3956127 * r3956128;
        double r3956130 = r3956126 + r3956129;
        double r3956131 = 1.0;
        double r3956132 = 0.99229;
        double r3956133 = 0.04481;
        double r3956134 = r3956127 * r3956133;
        double r3956135 = r3956132 + r3956134;
        double r3956136 = r3956127 * r3956135;
        double r3956137 = r3956131 + r3956136;
        double r3956138 = r3956130 / r3956137;
        double r3956139 = r3956138 - r3956127;
        return r3956139;
}

↓

double f(double x) {
        double r3956140 = 0.27061;
        double r3956141 = x;
        double r3956142 = r3956140 * r3956141;
        double r3956143 = 2.30753;
        double r3956144 = r3956142 + r3956143;
        double r3956145 = 0.04481;
        double r3956146 = r3956145 * r3956141;
        double r3956147 = 0.99229;
        double r3956148 = r3956146 + r3956147;
        double r3956149 = r3956141 * r3956148;
        double r3956150 = r3956149 * r3956149;
        double r3956151 = r3956149 * r3956150;
        double r3956152 = cbrt(r3956151);
        double r3956153 = 1.0;
        double r3956154 = r3956152 + r3956153;
        double r3956155 = r3956144 / r3956154;
        double r3956156 = r3956155 * r3956155;
        double r3956157 = r3956155 * r3956156;
        double r3956158 = cbrt(r3956157);
        double r3956159 = r3956158 - r3956141;
        return r3956159;
}

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-cbrt-cube 0.0

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

4. Applied add-cbrt-cube 0.0

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

5. Applied cbrt-unprod 0.0

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

6. Simplified 0.0

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

8. Applied add-cbrt-cube 0.0

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

9. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019170 
(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))