Average Error: 23.6 → 11.9
Time: 5.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.657257095770849 \cdot 10^{+96} \lor \neg \left(\alpha \leq 1.7621934558652523 \cdot 10^{+226}\right) \land \alpha \leq 3.9128100197110995 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{1}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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 \leq 2.657257095770849 \cdot 10^{+96} \lor \neg \left(\alpha \leq 1.7621934558652523 \cdot 10^{+226}\right) \land \alpha \leq 3.9128100197110995 \cdot 10^{+277}:\\
\;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{1}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right) + 1}{2}\\

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

\end{array}
double code(double alpha, double beta, double i) {
	return (((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 <= 2.657257095770849e+96) || (!(alpha <= 1.7621934558652523e+226) && (alpha <= 3.9128100197110995e+277)))) {
		VAR = (((double) (((double) (((double) (alpha + beta)) * ((double) ((1.0 / ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))) * (((double) (beta - alpha)) / ((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))))) + 1.0)) / 2.0);
	} else {
		VAR = (((double) ((2.0 / alpha) + ((double) ((8.0 / ((double) pow(alpha, 3.0))) - (4.0 / ((double) (alpha * alpha))))))) / 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 < 2.6572570957708492e96 or 1.7621934558652523e226 < alpha < 3.9128100197110995e277

    1. Initial program Error: 16.7 bits

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

    2. SimplifiedError: 13.4 bits

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

    4. Applied *-un-lft-identityError: 13.4 bits

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

    5. Applied times-fracError: 5.9 bits

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

    6. SimplifiedError: 5.9 bits

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

    7. SimplifiedError: 5.9 bits

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

    if 2.6572570957708492e96 < alpha < 1.7621934558652523e226 or 3.9128100197110995e277 < alpha

    1. Initial program Error: 56.5 bits

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

    2. SimplifiedError: 48.3 bits

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

    3. Taylor expanded around inf Error: 40.7 bits

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

    4. SimplifiedError: 40.7 bits

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

  4. Final simplificationError: 11.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.657257095770849 \cdot 10^{+96} \lor \neg \left(\alpha \leq 1.7621934558652523 \cdot 10^{+226}\right) \land \alpha \leq 3.9128100197110995 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{1}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))