Average Error: 23.3 → 12.1
Time: 39.1s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.958204727471386 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 2.1817558498141728 \cdot 10^{+73} \lor \neg \left(\alpha \leq 8.457234180095517 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\sqrt{\alpha + \beta} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt{\alpha + \beta}\right)}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]
\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 7.958204727471386 \cdot 10^{+26}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}{2}\\

\mathbf{elif}\;\alpha \leq 2.1817558498141728 \cdot 10^{+73} \lor \neg \left(\alpha \leq 8.457234180095517 \cdot 10^{+129}\right):\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \frac{\sqrt{\alpha + \beta} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt{\alpha + \beta}\right)}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\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 7.958204727471386e+26)
   (/
    (cbrt
     (pow
      (+
       (*
        (+ alpha beta)
        (/
         (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
         (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
       1.0)
      3.0))
    2.0)
   (if (or (<= alpha 2.1817558498141728e+73)
           (not (<= alpha 8.457234180095517e+129)))
     (/
      (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
      2.0)
     (/
      (+
       1.0
       (/
        (*
         (sqrt (+ alpha beta))
         (*
          (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
          (sqrt (+ alpha beta))))
        (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
      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 <= 7.958204727471386e+26) {
		tmp = cbrt(pow((((alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i))))) + 1.0), 3.0)) / 2.0;
	} else if ((alpha <= 2.1817558498141728e+73) || !(alpha <= 8.457234180095517e+129)) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = (1.0 + ((sqrt(alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) * sqrt(alpha + beta))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 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 3 regimes

  2. if alpha < 7.95820472747138644e26

    1. Initial program 10.5

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

    3. Applied add-sqr-sqrt_binary6410.5

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

    4. Applied *-un-lft-identity_binary6410.5

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

    5. Applied times-frac_binary640.5

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

    6. Applied times-frac_binary640.5

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

    7. Simplified0.5

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

    8. Simplified0.5

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

    10. Applied div-inv_binary640.6

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

    11. Applied associate-*l*_binary640.5

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

    12. Simplified0.5

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

    14. Applied add-cbrt-cube_binary640.5

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

    15. Simplified0.5

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

    if 7.95820472747138644e26 < alpha < 2.1817558498141728e73 or 8.4572341800955171e129 < alpha

    1. Initial program 55.9

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

    2. Taylor expanded around inf 41.1

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

    3. Simplified41.1

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

    if 2.1817558498141728e73 < alpha < 8.4572341800955171e129

    1. Initial program 41.0

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

    3. Applied *-un-lft-identity_binary6441.0

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

    4. Applied times-frac_binary6429.2

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

    5. Simplified29.2

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

    7. Applied add-sqr-sqrt_binary6429.2

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

    8. Applied associate-*l*_binary6429.2

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

    9. Simplified29.2

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.958204727471386 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 2.1817558498141728 \cdot 10^{+73} \lor \neg \left(\alpha \leq 8.457234180095517 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\sqrt{\alpha + \beta} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt{\alpha + \beta}\right)}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020263 
(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))