Average Error: 0.0 → 0.1
Time: 14.6s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\sqrt{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}} \cdot \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{\sqrt{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{1}{\sqrt{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}} \cdot \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{\sqrt{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}}
double f(double x) {
        double r85457 = x;
        double r85458 = 2.30753;
        double r85459 = 0.27061;
        double r85460 = r85457 * r85459;
        double r85461 = r85458 + r85460;
        double r85462 = 1.0;
        double r85463 = 0.99229;
        double r85464 = 0.04481;
        double r85465 = r85457 * r85464;
        double r85466 = r85463 + r85465;
        double r85467 = r85466 * r85457;
        double r85468 = r85462 + r85467;
        double r85469 = r85461 / r85468;
        double r85470 = r85457 - r85469;
        return r85470;
}
↓
double f(double x) {
        double r85471 = x;
        double r85472 = 1.0;
        double r85473 = 1.0;
        double r85474 = 0.99229;
        double r85475 = 0.04481;
        double r85476 = r85471 * r85475;
        double r85477 = r85474 + r85476;
        double r85478 = r85477 * r85471;
        double r85479 = r85473 + r85478;
        double r85480 = sqrt(r85479);
        double r85481 = r85472 / r85480;
        double r85482 = 2.30753;
        double r85483 = 0.27061;
        double r85484 = r85471 * r85483;
        double r85485 = r85482 + r85484;
        double r85486 = r85485 / r85480;
        double r85487 = r85481 * r85486;
        double r85488 = r85471 - r85487;
        return r85488;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqr-sqrt0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied times-frac0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* (+ 0.99229 (* x 0.04481)) x)))))