Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D

Average Error: 0.0 → 0.0

Time: 20.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r5616451 = x;
        double r5616452 = 2.30753;
        double r5616453 = 0.27061;
        double r5616454 = r5616451 * r5616453;
        double r5616455 = r5616452 + r5616454;
        double r5616456 = 1.0;
        double r5616457 = 0.99229;
        double r5616458 = 0.04481;
        double r5616459 = r5616451 * r5616458;
        double r5616460 = r5616457 + r5616459;
        double r5616461 = r5616460 * r5616451;
        double r5616462 = r5616456 + r5616461;
        double r5616463 = r5616455 / r5616462;
        double r5616464 = r5616451 - r5616463;
        return r5616464;
}

↓

double f(double x) {
        double r5616465 = x;
        double r5616466 = 2.30753;
        double r5616467 = 0.27061;
        double r5616468 = r5616465 * r5616467;
        double r5616469 = r5616466 + r5616468;
        double r5616470 = 0.04481;
        double r5616471 = r5616465 * r5616470;
        double r5616472 = 0.99229;
        double r5616473 = r5616471 + r5616472;
        double r5616474 = r5616473 * r5616465;
        double r5616475 = r5616474 * r5616474;
        double r5616476 = r5616474 * r5616475;
        double r5616477 = cbrt(r5616476);
        double r5616478 = 1.0;
        double r5616479 = r5616477 + r5616478;
        double r5616480 = r5616469 / r5616479;
        double r5616481 = r5616480 * r5616480;
        double r5616482 = r5616480 * r5616481;
        double r5616483 = cbrt(r5616482);
        double r5616484 = r5616465 - r5616483;
        return r5616484;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-cbrt-cube 0.0

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

4. Applied add-cbrt-cube 0.0

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

5. Applied cbrt-unprod 0.0

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

6. Simplified 0.0

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

8. Applied add-cbrt-cube 0.0

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

9. Final simplification 0.0

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

Reproduce

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