Average Error: 23.2 → 11.1
Time: 5.6s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.0852138065724436 \cdot 10^{+122}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)} + 1}}\right) + \left(\log \left(\sqrt{{\left(e^{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)}}\right)}^{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right)}}\right) + \log \left(\sqrt{e^{1}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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 2.0852138065724436 \cdot 10^{+122}:\\
\;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)} + 1}}\right) + \left(\log \left(\sqrt{{\left(e^{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)}}\right)}^{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right)}}\right) + \log \left(\sqrt{e^{1}}\right)\right)}{2}\\

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

\end{array}
(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))
↓
(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.0852138065724436e+122)
   (/
    (+
     (log
      (sqrt
       (exp
        (+
         (*
          (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i))))
          (/ (- beta alpha) (+ beta (+ alpha (+ 2.0 (* 2.0 i))))))
         1.0))))
     (+
      (log
       (sqrt
        (pow
         (exp (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i)))))
         (/ (- beta alpha) (+ beta (+ alpha (+ 2.0 (* 2.0 i))))))))
      (log (sqrt (exp 1.0)))))
    2.0)
   (/
    (+ (/ 2.0 alpha) (- (/ 8.0 (pow alpha 3.0)) (/ 4.0 (* alpha alpha))))
    2.0)))
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 tmp;
	if ((alpha <= 2.0852138065724436e+122)) {
		tmp = (((double) (((double) log(((double) sqrt(((double) exp(((double) (((double) ((((double) (alpha + beta)) / ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))) * (((double) (beta - alpha)) / ((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))) + 1.0)))))))) + ((double) (((double) log(((double) sqrt(((double) pow(((double) exp((((double) (alpha + beta)) / ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))))), (((double) (beta - alpha)) / ((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))))))) + ((double) log(((double) sqrt(((double) exp(1.0)))))))))) / 2.0);
	} else {
		tmp = (((double) ((2.0 / alpha) + ((double) ((8.0 / ((double) pow(alpha, 3.0))) - (4.0 / ((double) (alpha * alpha))))))) / 2.0);
	}
	return tmp;
}

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 < 2.08521380657244364e122

    1. Initial program Error: 13.9 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: 11.1 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: 11.1 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: 3.7 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. Applied associate-*r*Error: 3.7 bits

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

    7. SimplifiedError: 3.7 bits

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

    9. Applied add-log-expError: 3.7 bits

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

    10. Applied add-log-expError: 3.7 bits

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

    11. Applied sum-logError: 3.7 bits

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

    12. SimplifiedError: 3.7 bits

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

    14. Applied add-sqr-sqrtError: 3.7 bits

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

    15. Applied log-prodError: 3.7 bits

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

    17. Applied exp-sumError: 3.7 bits

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

    18. Applied sqrt-prodError: 3.7 bits

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

    19. Applied log-prodError: 3.7 bits

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

    20. SimplifiedError: 3.7 bits

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

    if 2.08521380657244364e122 < alpha

    1. Initial program Error: 61.2 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: 51.6 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: 41.5 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: 41.5 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: 11.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.0852138065724436 \cdot 10^{+122}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)} + 1}}\right) + \left(\log \left(\sqrt{{\left(e^{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)}}\right)}^{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right)}}\right) + \log \left(\sqrt{e^{1}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))