Average Error: 23.5 → 7.4
Time: 10.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -0.99999999999995937:\\ \;\;\;\;\frac{8 \cdot \frac{1}{{\alpha}^{3}} - \left(\frac{\frac{4}{\alpha}}{\alpha} - 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1}{2}\\ \end{array}\]
\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}\;\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} \le -0.99999999999995937:\\
\;\;\;\;\frac{8 \cdot \frac{1}{{\alpha}^{3}} - \left(\frac{\frac{4}{\alpha}}{\alpha} - 2 \cdot \frac{1}{\alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1}{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 ((((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)))) <= -0.9999999999999594)) {
		VAR = ((double) (((double) (((double) (8.0 * ((double) (1.0 / ((double) pow(alpha, 3.0)))))) - ((double) (((double) (((double) (4.0 / alpha)) / alpha)) - ((double) (2.0 * ((double) (1.0 / alpha)))))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) (alpha + beta)) / 1.0)) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)) / ((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))))))) + 1.0)) / 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 beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) < -0.9999999999999594

    1. Initial program 63.1

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

    3. Applied *-un-lft-identity63.1

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

    4. Applied times-frac54.1

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

    5. Applied associate-/l*54.1

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

    7. Applied add-cube-cbrt54.5

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

    8. Applied *-un-lft-identity54.5

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

    9. Applied times-frac54.4

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

    10. Applied *-un-lft-identity54.4

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

    11. Applied times-frac54.5

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

    12. Applied add-cube-cbrt54.3

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

    13. Applied times-frac54.3

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

    14. Simplified54.2

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

    15. Simplified54.1

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

    16. Taylor expanded around inf 32.7

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

    17. Simplified32.7

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

    if -0.9999999999999594 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))

    1. Initial program 12.4

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

    3. Applied *-un-lft-identity12.4

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

    4. Applied times-frac0.2

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

    5. Applied associate-/l*0.2

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -0.99999999999995937:\\ \;\;\;\;\frac{8 \cdot \frac{1}{{\alpha}^{3}} - \left(\frac{\frac{4}{\alpha}}{\alpha} - 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(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))