Average Error: 0.0 → 0.0
Time: 15.5s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}
double f(double x) {
        double r59432 = x;
        double r59433 = 2.30753;
        double r59434 = 0.27061;
        double r59435 = r59432 * r59434;
        double r59436 = r59433 + r59435;
        double r59437 = 1.0;
        double r59438 = 0.99229;
        double r59439 = 0.04481;
        double r59440 = r59432 * r59439;
        double r59441 = r59438 + r59440;
        double r59442 = r59441 * r59432;
        double r59443 = r59437 + r59442;
        double r59444 = r59436 / r59443;
        double r59445 = r59432 - r59444;
        return r59445;
}
↓
double f(double x) {
        double r59446 = x;
        double r59447 = 0.27061;
        double r59448 = 2.30753;
        double r59449 = fma(r59446, r59447, r59448);
        double r59450 = 1.0;
        double r59451 = 0.04481;
        double r59452 = 0.99229;
        double r59453 = fma(r59451, r59446, r59452);
        double r59454 = 1.0;
        double r59455 = fma(r59453, r59446, r59454);
        double r59456 = r59450 / r59455;
        double r59457 = r59449 * r59456;
        double r59458 = r59446 - r59457;
        return r59458;
}

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. Simplified0.0

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

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061000000000002)) (+ 1 (* (+ 0.992290000000000005 (* x 0.044810000000000003)) x)))))