Average Error: 0.0 → 0.0
Time: 14.2s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)\right)}^{3}}} - x\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)\right)}^{3}}} - x
double f(double x) {
        double r56488 = 2.30753;
        double r56489 = x;
        double r56490 = 0.27061;
        double r56491 = r56489 * r56490;
        double r56492 = r56488 + r56491;
        double r56493 = 1.0;
        double r56494 = 0.99229;
        double r56495 = 0.04481;
        double r56496 = r56489 * r56495;
        double r56497 = r56494 + r56496;
        double r56498 = r56489 * r56497;
        double r56499 = r56493 + r56498;
        double r56500 = r56492 / r56499;
        double r56501 = r56500 - r56489;
        return r56501;
}
↓
double f(double x) {
        double r56502 = x;
        double r56503 = 0.27061;
        double r56504 = 2.30753;
        double r56505 = fma(r56502, r56503, r56504);
        double r56506 = 0.04481;
        double r56507 = 0.99229;
        double r56508 = fma(r56506, r56502, r56507);
        double r56509 = 1.0;
        double r56510 = fma(r56508, r56502, r56509);
        double r56511 = 3.0;
        double r56512 = pow(r56510, r56511);
        double r56513 = cbrt(r56512);
        double r56514 = r56505 / r56513;
        double r56515 = r56514 - r56502;
        return r56515;
}

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. Simplified0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061000000000002)) (+ 1 (* x (+ 0.992290000000000005 (* x 0.044810000000000003))))) x))