Average Error: 24.1 → 11.6
Time: 9.4s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.86408341367568 \cdot 10^{+222}:\\ \;\;\;\;\frac{1 + \frac{1}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 5.86408341367568 \cdot 10^{+222}:\\
\;\;\;\;\frac{1 + \frac{1}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{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 5.86408341367568e+222)
   (/
    (+
     1.0
     (*
      (/
       1.0
       (*
        (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
        (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
      (*
       (+ alpha beta)
       (/
        (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
        (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))))
    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 (((((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 tmp;
	if (alpha <= 5.86408341367568e+222) {
		tmp = (1.0 + ((1.0 / (cbrt(2.0 + ((alpha + beta) + (2.0 * i))) * cbrt(2.0 + ((alpha + beta) + (2.0 * i))))) * ((alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / cbrt(2.0 + ((alpha + beta) + (2.0 * i))))))) / 2.0;
	} else {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (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 < 5.8640834136756799e222

    1. Initial program 19.9

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

    3. Applied add-cube-cbrt_binary6420.0

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

    4. Applied *-un-lft-identity_binary6420.0

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

    5. Applied times-frac_binary6420.0

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

    6. Simplified20.0

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

    7. Simplified8.4

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

    if 5.8640834136756799e222 < alpha

    1. Initial program 64.0

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

    2. Taylor expanded around inf 42.0

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

    3. Simplified42.0

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.86408341367568 \cdot 10^{+222}:\\ \;\;\;\;\frac{1 + \frac{1}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020260 
(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))