Average Error: 24.2 → 11.5
Time: 11.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.3286644476139376 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{\frac{{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3} + {\left({1}^{3}\right)}^{3}}{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} + {1}^{6}\right) - {\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3} \cdot {1}^{3}}}{1 \cdot \left(1 - \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right) + \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 7.3286644476139376 \cdot 10^{139}:\\
\;\;\;\;\frac{\frac{\frac{{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3} + {\left({1}^{3}\right)}^{3}}{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} + {1}^{6}\right) - {\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3} \cdot {1}^{3}}}{1 \cdot \left(1 - \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right) + \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\

\end{array}
double code(double alpha, double beta, double i) {
	return ((((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 7.328664447613938e+139)) {
		VAR = ((((pow(pow((1.0 / ((((alpha + beta) + (2.0 * i)) + 2.0) / ((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))), 3.0), 3.0) + pow(pow(1.0, 3.0), 3.0)) / ((pow((1.0 / ((((alpha + beta) + (2.0 * i)) + 2.0) / ((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))), 6.0) + pow(1.0, 6.0)) - (pow((1.0 / ((((alpha + beta) + (2.0 * i)) + 2.0) / ((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))), 3.0) * pow(1.0, 3.0)))) / ((1.0 * (1.0 - (1.0 / ((((alpha + beta) + (2.0 * i)) + 2.0) / ((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))))) + ((1.0 / ((((alpha + beta) + (2.0 * i)) + 2.0) / ((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))) * (1.0 / ((((alpha + beta) + (2.0 * i)) + 2.0) / ((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i))))))))) / 2.0);
	} else {
		VAR = ((((2.0 * (1.0 / alpha)) + (8.0 * (1.0 / pow(alpha, 3.0)))) - (4.0 * (1.0 / 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 < 7.328664447613938e+139

    1. Initial program 15.6

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

    3. Applied *-un-lft-identity15.6

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

    4. Applied times-frac4.8

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

    5. Simplified4.8

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

    7. Applied clear-num4.8

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

    9. Applied flip3-+4.8

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

    10. Simplified4.8

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

    12. Applied flip3-+4.8

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

    13. Simplified4.8

      \[\leadsto \frac{\frac{\frac{{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3} + {\left({1}^{3}\right)}^{3}}{\color{blue}{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} + {1}^{6}\right) - {\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3} \cdot {1}^{3}}}}{1 \cdot \left(1 - \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right) + \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{2}\]

    if 7.328664447613938e+139 < alpha

    1. Initial program 62.5

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

    3. Applied *-un-lft-identity62.5

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

    4. Applied times-frac46.8

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

    5. Simplified46.8

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

    7. Applied clear-num46.8

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

    8. Taylor expanded around inf 41.9

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.3286644476139376 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{\frac{{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3} + {\left({1}^{3}\right)}^{3}}{\left({\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} + {1}^{6}\right) - {\left(\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3} \cdot {1}^{3}}}{1 \cdot \left(1 - \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right) + \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))