Average Error: 23.9 → 13.7
Time: 6.3s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot i \leq 4.3536151651021257 \cdot 10^{-219} \lor \neg \left(2 \cdot i \leq 1.0861756241842032 \cdot 10^{-199}\right):\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(2 \cdot i + \beta\right)}{\beta - \alpha} \cdot \sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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}\;2 \cdot i \leq 4.3536151651021257 \cdot 10^{-219} \lor \neg \left(2 \cdot i \leq 1.0861756241842032 \cdot 10^{-199}\right):\\
\;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(2 \cdot i + \beta\right)}{\beta - \alpha} \cdot \sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{\alpha + \left(\beta + \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}
(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 (or (<= (* 2.0 i) 4.3536151651021257e-219)
         (not (<= (* 2.0 i) 1.0861756241842032e-199)))
   (/
    (+
     (*
      (+ alpha beta)
      (*
       (*
        (cbrt 1.0)
        (/
         (cbrt 1.0)
         (*
          (/ (+ alpha (+ (* 2.0 i) beta)) (- beta alpha))
          (sqrt (+ alpha (+ beta (+ 2.0 (* 2.0 i))))))))
       (/ (cbrt 1.0) (sqrt (+ alpha (+ beta (+ 2.0 (* 2.0 i))))))))
     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 VAR;
	if (((((double) (2.0 * i)) <= 4.3536151651021257e-219) || !(((double) (2.0 * i)) <= 1.0861756241842032e-199))) {
		VAR = (((double) (((double) (((double) (alpha + beta)) * ((double) (((double) (((double) cbrt(1.0)) * (((double) cbrt(1.0)) / ((double) ((((double) (alpha + ((double) (((double) (2.0 * i)) + beta)))) / ((double) (beta - alpha))) * ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i))))))))))))))) * (((double) cbrt(1.0)) / ((double) sqrt(((double) (alpha + ((double) (beta + ((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 (* 2.0 i) < 4.35361516510212565e-219 or 1.08617562418420315e-199 < (* 2.0 i)

    1. Initial program 23.9

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

    2. Simplified19.5

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

    4. Applied clear-num19.5

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

    5. Simplified12.4

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

    7. Applied add-sqr-sqrt12.5

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

    8. Applied associate-*r*12.5

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

    9. Simplified12.5

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

    11. Applied add-cube-cbrt12.5

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

    12. Applied times-frac12.5

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

    13. Simplified12.5

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

    14. Simplified12.5

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

    if 4.35361516510212565e-219 < (* 2.0 i) < 1.08617562418420315e-199

    1. Initial program 23.2

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

    2. Simplified21.4

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

    3. Taylor expanded around inf 50.3

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

    4. Simplified50.3

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

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 4.3536151651021257 \cdot 10^{-219} \lor \neg \left(2 \cdot i \leq 1.0861756241842032 \cdot 10^{-199}\right):\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(2 \cdot i + \beta\right)}{\beta - \alpha} \cdot \sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))