Average Error: 23.7 → 11.0
Time: 6.7s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 834912956515337220:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}\right)}{2}\\ \mathbf{elif}\;\alpha \le 4.10784956866542257 \cdot 10^{28} \lor \neg \left(\alpha \le 3.3101337033498082 \cdot 10^{173}\right):\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\sqrt{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} - \sqrt{1}\right)}{2}\\ \end{array}\]
\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 834912956515337220:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}\right)}{2}\\

\mathbf{elif}\;\alpha \le 4.10784956866542257 \cdot 10^{28} \lor \neg \left(\alpha \le 3.3101337033498082 \cdot 10^{173}\right):\\
\;\;\;\;\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}\\

\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(\sqrt{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} - \sqrt{1}\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 <= 8.349129565153372e+17)) {
		VAR = ((double) (((double) log(((double) exp(((double) (((double) (((double) (beta - alpha)) / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))))))) + 1.0)))))) / 2.0));
	} else {
		double VAR_1;
		if (((alpha <= 4.107849568665423e+28) || !(alpha <= 3.310133703349808e+173))) {
			VAR_1 = ((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));
		} else {
			VAR_1 = ((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) sqrt(((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) sqrt(1.0)))) * ((double) (((double) sqrt(((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) sqrt(1.0)))))))) / 2.0));
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if alpha < 834912956515337220

    1. Initial program 11.0

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

    2. Simplified0.4

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

    4. Applied add-log-exp0.4

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

    5. Applied add-log-exp0.4

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

    6. Applied sum-log0.4

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

    7. Simplified0.4

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

    if 834912956515337220 < alpha < 4.10784956866542257e28 or 3.3101337033498082e173 < alpha

    1. Initial program 61.6

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

    2. Simplified45.7

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

    4. Applied div-sub45.7

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

    5. Applied associate-+l-44.1

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

    6. Simplified44.1

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

    7. Taylor expanded around inf 39.1

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

    8. Simplified39.1

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

    if 4.10784956866542257e28 < alpha < 3.3101337033498082e173

    1. Initial program 42.1

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

    2. Simplified30.6

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

    4. Applied div-sub30.6

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

    5. Applied associate-+l-30.0

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

    6. Simplified30.0

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

    8. Applied add-sqr-sqrt30.0

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

    9. Applied add-sqr-sqrt30.0

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

    10. Applied difference-of-squares30.0

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

    11. Simplified30.0

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

    12. Simplified30.0

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 834912956515337220:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}\right)}{2}\\ \mathbf{elif}\;\alpha \le 4.10784956866542257 \cdot 10^{28} \lor \neg \left(\alpha \le 3.3101337033498082 \cdot 10^{173}\right):\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\sqrt{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} - \sqrt{1}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))