Average Error: 24.1 → 11.8
Time: 7.4s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.5251724157134815 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 1.0405905765331085 \cdot 10^{+86} \lor \neg \left(\alpha \leq 1.3588989434573756 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)}^{3}}}{2}\\ \end{array}\]
\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 1.5251724157134815 \cdot 10^{+41}:\\
\;\;\;\;\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 1.0405905765331085 \cdot 10^{+86} \lor \neg \left(\alpha \leq 1.3588989434573756 \cdot 10^{+212}\right):\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)}^{3}}}{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 1.5251724157134815e+41)
   (/
    (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 1.0405905765331085e+86)
           (not (<= alpha 1.3588989434573756e+212)))
     (/
      (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
      2.0)
     (/
      (cbrt
       (pow
        (+
         1.0
         (*
          (/ (+ alpha beta) (sqrt (+ (+ alpha beta) (* 2.0 i))))
          (/
           (/ (- beta alpha) (sqrt (+ (+ alpha beta) (* 2.0 i))))
           (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
        3.0))
      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 <= 1.5251724157134815e+41) {
		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 <= 1.0405905765331085e+86) || !(alpha <= 1.3588989434573756e+212)) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = cbrt(pow((1.0 + (((alpha + beta) / sqrt((alpha + beta) + (2.0 * i))) * (((beta - alpha) / sqrt((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i)))))), 3.0)) / 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 < 1.5251724157134815e41

    1. Initial program 11.6

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

    3. Applied *-un-lft-identity_binary6411.6

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

    4. Applied *-un-lft-identity_binary6411.6

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

    5. Applied times-frac_binary640.8

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

    6. Applied times-frac_binary640.8

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

    7. Simplified0.8

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

    8. Simplified0.8

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

    10. Applied add-cbrt-cube_binary640.9

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

    11. Simplified0.9

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

    if 1.5251724157134815e41 < alpha < 1.0405905765331085e86 or 1.3588989434573756e212 < alpha

    1. Initial program 55.6

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

    2. Taylor expanded around inf 41.2

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

    3. Simplified41.2

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

    if 1.0405905765331085e86 < alpha < 1.3588989434573756e212

    1. Initial program 53.1

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

    3. Applied *-un-lft-identity_binary6453.1

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

    4. Applied *-un-lft-identity_binary6453.1

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

    5. Applied times-frac_binary6435.4

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

    6. Applied times-frac_binary6435.3

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

    7. Simplified35.3

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

    8. Simplified35.3

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

    10. Applied add-cbrt-cube_binary6435.3

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

    11. Simplified35.3

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

    13. Applied *-un-lft-identity_binary6435.3

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

    14. Applied add-sqr-sqrt_binary6435.4

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

    15. Applied *-un-lft-identity_binary6435.4

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

    16. Applied times-frac_binary6435.3

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

    17. Applied times-frac_binary6435.3

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

    18. Applied associate-*r*_binary6435.4

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

    19. Simplified35.4

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.5251724157134815 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 1.0405905765331085 \cdot 10^{+86} \lor \neg \left(\alpha \leq 1.3588989434573756 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))