Average Error: 24.4 → 12.5
Time: 13.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.77904391480143721 \cdot 10^{77} \lor \neg \left(\alpha \le 7.9741769007242744 \cdot 10^{226} \lor \neg \left(\alpha \le 4.15900563899299625 \cdot 10^{257}\right)\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\alpha + \beta} \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 2.77904391480143721 \cdot 10^{77} \lor \neg \left(\alpha \le 7.9741769007242744 \cdot 10^{226} \lor \neg \left(\alpha \le 4.15900563899299625 \cdot 10^{257}\right)\right):\\
\;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\alpha + \beta} \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\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 <= 2.779043914801437e+77) || !((alpha <= 7.974176900724274e+226) || !(alpha <= 4.159005638992996e+257)))) {
		VAR = ((double) (((double) cbrt(((double) pow(((double) fma(((double) (((double) sqrt(1.0)) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)))), ((double) (((double) sqrt(1.0)) / ((double) (((double) (1.0 / ((double) (alpha + beta)))) * ((double) (((double) fma(i, 2.0, ((double) (alpha + beta)))) / ((double) (beta - alpha)))))))), 1.0)), 3.0)))) / 2.0));
	} else {
		VAR = ((double) (((double) fma(2.0, ((double) (1.0 / alpha)), ((double) (((double) (8.0 * ((double) (1.0 / ((double) pow(alpha, 3.0)))))) - ((double) (4.0 * ((double) (1.0 / ((double) pow(alpha, 2.0)))))))))) / 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 < 2.779043914801437e+77 or 7.974176900724274e+226 < alpha < 4.159005638992996e+257

    1. Initial program 15.5

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

    3. Applied clear-num15.5

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

    4. Simplified4.6

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

    6. Applied div-inv4.6

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

    7. Applied associate-*l*4.6

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

    8. Applied add-sqr-sqrt4.6

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

    9. Applied times-frac4.5

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

    10. Applied fma-def4.5

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

    12. Applied add-cbrt-cube4.6

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

    13. Simplified4.6

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

    if 2.779043914801437e+77 < alpha < 7.974176900724274e+226 or 4.159005638992996e+257 < alpha

    1. Initial program 55.7

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

    2. Taylor expanded around inf 40.2

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

    3. Simplified40.2

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.77904391480143721 \cdot 10^{77} \lor \neg \left(\alpha \le 7.9741769007242744 \cdot 10^{226} \lor \neg \left(\alpha \le 4.15900563899299625 \cdot 10^{257}\right)\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\alpha + \beta} \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))