Average Error: 54.3 → 37.2
Time: 22.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.8523098755484387 \cdot 10^{210}:\\ \;\;\;\;\frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.25, i, 0.25 \cdot \beta\right)\right)\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.8523098755484387 \cdot 10^{210}:\\
\;\;\;\;\frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.25, i, 0.25 \cdot \beta\right)\right)\\

\end{array}
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (i * ((double) (((double) (alpha + beta)) + i)))) * ((double) (((double) (beta * alpha)) + ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))))) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) - 1.0))));
}
↓
double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 1.8523098755484387e+210)) {
		VAR = ((double) (((double) (((double) (((double) (i / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + ((double) sqrt(1.0)))))) * ((double) (((double) (alpha + beta)) + i)))) / ((double) (((double) cbrt(((double) fma(i, 2.0, ((double) (alpha + beta)))))) * ((double) cbrt(((double) fma(i, 2.0, ((double) (alpha + beta)))))))))) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) - ((double) sqrt(1.0)))) * ((double) (((double) (((double) (((double) cbrt(((double) sqrt(((double) fma(i, 2.0, ((double) (alpha + beta)))))))) / ((double) (((double) cbrt(((double) fma(beta, alpha, ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))) * ((double) sqrt(((double) cbrt(((double) fma(beta, alpha, ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))))))))) * ((double) (((double) (((double) cbrt(((double) sqrt(((double) fma(i, 2.0, ((double) (alpha + beta)))))))) * ((double) sqrt(((double) fma(i, 2.0, ((double) (alpha + beta)))))))) / ((double) sqrt(((double) cbrt(((double) fma(beta, alpha, ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))))))))) * ((double) (((double) sqrt(((double) fma(i, 2.0, ((double) (alpha + beta)))))) / ((double) cbrt(((double) fma(beta, alpha, ((double) (i * ((double) (((double) (alpha + beta)) + i))))))))))))))));
	} else {
		VAR = ((double) (((double) (i / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) - 1.0)))) * ((double) fma(0.25, alpha, ((double) fma(0.25, i, ((double) (0.25 * beta))))))));
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.8523098755484387e+210

    1. Initial program 53.1

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

    2. Simplified52.4

      \[\leadsto \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity52.4

      \[\leadsto \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    5. Applied associate-*l*52.4

      \[\leadsto \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    6. Applied times-frac39.7

      \[\leadsto \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}{1} \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    7. Applied times-frac38.4

      \[\leadsto \color{blue}{\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    8. Simplified38.4

      \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity38.4

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    11. Applied add-cube-cbrt38.8

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    12. Applied associate-*l*38.8

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    13. Applied times-frac38.8

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\alpha + \beta\right) + i}{\color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    14. Applied associate-/r*38.8

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \color{blue}{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1}}}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    15. Applied add-sqr-sqrt38.8

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1} \cdot \sqrt{1}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1}}}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    16. Applied difference-of-squares38.8

      \[\leadsto \frac{i}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1}}}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    17. Applied associate-/r*36.9

      \[\leadsto \color{blue}{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1}}}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    18. Applied frac-times36.3

      \[\leadsto \color{blue}{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    19. Simplified36.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]
    20. Using strategy rm

    21. Applied add-cube-cbrt36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}\]

    22. Applied add-sqr-sqrt36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    23. Applied associate-*r*36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    24. Applied times-frac36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}}\]
    25. Using strategy rm

    26. Applied add-sqr-sqrt36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\]

    27. Applied associate-*r*36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\frac{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right) \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\]

    28. Applied add-sqr-sqrt36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\frac{\sqrt[3]{\color{blue}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right) \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\]

    29. Applied cbrt-prod36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\right)} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right) \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\]

    30. Applied associate-*l*36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\frac{\color{blue}{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}}{\left(\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right) \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\]

    31. Applied times-frac36.4

      \[\leadsto \frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\]

    if 1.8523098755484387e+210 < alpha

    1. Initial program 64.0

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

    2. Simplified57.8

      \[\leadsto \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity57.8

      \[\leadsto \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    5. Applied associate-*l*57.8

      \[\leadsto \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}{1 \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    6. Applied times-frac57.8

      \[\leadsto \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}{1} \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    7. Applied times-frac57.8

      \[\leadsto \color{blue}{\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\]

    8. Simplified57.8

      \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    9. Taylor expanded around 0 43.3

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \color{blue}{\left(0.25 \cdot \alpha + \left(0.25 \cdot i + 0.25 \cdot \beta\right)\right)}\]

    10. Simplified43.3

      \[\leadsto \frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \color{blue}{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.25, i, 0.25 \cdot \beta\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification37.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.8523098755484387 \cdot 10^{210}:\\ \;\;\;\;\frac{\frac{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.25, i, 0.25 \cdot \beta\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))