Average Error: 23.7 → 12.5
Time: 17.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.479662422107554 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\sqrt{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 4.275223394131328 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\frac{\frac{\beta - \alpha}{\left|\sqrt[3]{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right|}}{\sqrt{\sqrt[3]{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}}{2}\\ \end{array}\]
\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 7.479662422107554 \cdot 10^{+100}:\\
\;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\sqrt{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} + 1}{2}\\

\mathbf{elif}\;\alpha \leq 4.275223394131328 \cdot 10^{+142}:\\
\;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\frac{\frac{\beta - \alpha}{\left|\sqrt[3]{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right|}}{\sqrt{\sqrt[3]{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\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 7.479662422107554e+100)
   (/
    (+
     (*
      (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i))))
      (/
       (/
        (/ (- beta alpha) (sqrt (sqrt (+ beta (+ alpha (+ 2.0 (* 2.0 i)))))))
        (sqrt (sqrt (+ beta (+ alpha (+ 2.0 (* 2.0 i)))))))
       (sqrt (+ alpha (+ beta (+ 2.0 (* 2.0 i)))))))
     1.0)
    2.0)
   (if (<= alpha 4.275223394131328e+142)
     (/
      (+ (/ 2.0 alpha) (- (/ 8.0 (pow alpha 3.0)) (/ 4.0 (* alpha alpha))))
      2.0)
     (/
      (+
       1.0
       (*
        (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i))))
        (/
         (/
          (/ (- beta alpha) (fabs (cbrt (+ beta (+ alpha (+ 2.0 (* 2.0 i)))))))
          (sqrt (cbrt (+ beta (+ alpha (+ 2.0 (* 2.0 i)))))))
         (sqrt (+ alpha (+ beta (+ 2.0 (* 2.0 i))))))))
      2.0))))
double code(double alpha, double beta, double i) {
	return (((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 tmp;
	if ((alpha <= 7.479662422107554e+100)) {
		tmp = (((double) (((double) ((((double) (alpha + beta)) / ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))) * (((((double) (beta - alpha)) / ((double) sqrt(((double) sqrt(((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))))) / ((double) sqrt(((double) sqrt(((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))))) / ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i))))))))))))) + 1.0)) / 2.0);
	} else {
		double tmp_1;
		if ((alpha <= 4.275223394131328e+142)) {
			tmp_1 = (((double) ((2.0 / alpha) + ((double) ((8.0 / ((double) pow(alpha, 3.0))) - (4.0 / ((double) (alpha * alpha))))))) / 2.0);
		} else {
			tmp_1 = (((double) (1.0 + ((double) ((((double) (alpha + beta)) / ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))) * (((((double) (beta - alpha)) / ((double) fabs(((double) cbrt(((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))))) / ((double) sqrt(((double) cbrt(((double) (beta + ((double) (alpha + ((double) (2.0 + ((double) (2.0 * i))))))))))))) / ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i))))))))))))))) / 2.0);
		}
		tmp = tmp_1;
	}
	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 < 7.4796624221075541e100

    1. Initial program Error: 13.8 bits

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

    2. SimplifiedError: 10.7 bits

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

    4. Applied *-un-lft-identityError: 10.7 bits

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

    5. Applied times-fracError: 3.2 bits

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

    6. Applied associate-*r*Error: 3.2 bits

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

    7. SimplifiedError: 3.2 bits

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

    9. Applied add-sqr-sqrtError: 3.2 bits

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

    10. Applied associate-/r*Error: 3.3 bits

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

    11. SimplifiedError: 3.3 bits

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

    13. Applied add-sqr-sqrtError: 3.3 bits

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

    14. Applied sqrt-prodError: 3.3 bits

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

    15. Applied associate-/r*Error: 3.3 bits

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

    if 7.4796624221075541e100 < alpha < 4.2752233941313279e142

    1. Initial program Error: 45.6 bits

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

    2. SimplifiedError: 42.2 bits

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

    3. Taylor expanded around inf Error: 41.5 bits

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

    4. SimplifiedError: 41.5 bits

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

    if 4.2752233941313279e142 < alpha

    1. Initial program Error: 62.7 bits

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

    2. SimplifiedError: 52.8 bits

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

    4. Applied *-un-lft-identityError: 52.8 bits

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

    5. Applied times-fracError: 46.0 bits

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

    6. Applied associate-*r*Error: 46.0 bits

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

    7. SimplifiedError: 46.0 bits

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

    9. Applied add-sqr-sqrtError: 46.4 bits

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

    10. Applied associate-/r*Error: 46.4 bits

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

    11. SimplifiedError: 46.4 bits

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

    13. Applied add-cube-cbrtError: 46.5 bits

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

    14. Applied sqrt-prodError: 46.5 bits

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

    15. Applied associate-/r*Error: 46.5 bits

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

    16. SimplifiedError: 46.5 bits

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

  4. Final simplificationError: 12.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.479662422107554 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\sqrt{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 4.275223394131328 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\frac{\frac{\beta - \alpha}{\left|\sqrt[3]{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right|}}{\sqrt{\sqrt[3]{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}}}{\sqrt{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))