Average Error: 24.1 → 10.9
Time: 6.8s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.72688726198891868 \cdot 10^{234}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \frac{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} \cdot \left(\alpha \cdot \frac{1}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}\right) - 1 \cdot 1}{1 + \left(\sqrt[3]{\alpha} \cdot \frac{\sqrt[3]{\alpha}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\sqrt[3]{\alpha}}{\alpha + \left(\beta + 2 \cdot i\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9.72688726198891868 \cdot 10^{234}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \frac{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} \cdot \left(\alpha \cdot \frac{1}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}\right) - 1 \cdot 1}{1 + \left(\sqrt[3]{\alpha} \cdot \frac{\sqrt[3]{\alpha}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\sqrt[3]{\alpha}}{\alpha + \left(\beta + 2 \cdot i\right)}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (alpha + beta)) * ((double) (beta - alpha)))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)))) + 1.0)) / 2.0));
}
↓
double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 9.726887261988919e+234)) {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))) - ((double) (((double) (((double) (((double) (alpha / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))) * ((double) (alpha * ((double) (1.0 / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))))))))))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + ((double) (((double) (((double) cbrt(alpha)) * ((double) (((double) cbrt(alpha)) / ((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)))))) * ((double) (((double) cbrt(alpha)) / ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))))))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))) - ((double) (((double) (4.0 / ((double) (alpha * alpha)))) - ((double) (((double) (2.0 / alpha)) + ((double) (8.0 / ((double) pow(alpha, 3.0)))))))))) / 2.0));
	}
	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 < 9.72688726198891868e234

    1. Initial program 20.8

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

    2. Simplified8.7

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

    4. Applied div-sub8.6

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

    5. Applied associate-+l-8.4

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

    6. Simplified8.4

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

    8. Applied flip--8.5

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

    9. Simplified8.5

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

    10. Simplified8.5

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

    12. Applied div-inv8.5

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

    13. Simplified8.5

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

    15. Applied add-cube-cbrt8.5

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

    16. Applied times-frac8.5

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

    17. Simplified8.5

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

    if 9.72688726198891868e234 < alpha

    1. Initial program 64.0

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

    2. Simplified52.1

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

    4. Applied div-sub52.1

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

    5. Applied associate-+l-50.1

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

    6. Simplified50.1

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

    7. Taylor expanded around inf 39.5

      \[\leadsto \frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    8. Simplified39.5

      \[\leadsto \frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.72688726198891868 \cdot 10^{234}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \frac{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} \cdot \left(\alpha \cdot \frac{1}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}\right) - 1 \cdot 1}{1 + \left(\sqrt[3]{\alpha} \cdot \frac{\sqrt[3]{\alpha}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\sqrt[3]{\alpha}}{\alpha + \left(\beta + 2 \cdot i\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))