Octave 3.8, jcobi/2

Average Error: 24.4 → 12.1

Time: 1.8min

Precision: binary64

\[\alpha > -1 \land \beta > -1 \land i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.158130970526724 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)}^{3}}\right)}{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - 1\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4.552813519470753 \cdot 10^{+166} \lor \neg \left(\alpha \leq 4.8907958261153134 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

FPCore

(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 7.158130970526724e+89)
   (/
    (/
     (log
      (exp
       (+
        1.0
        (pow
         (/
          (* (+ alpha beta) (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
          (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
         3.0))))
     (+
      1.0
      (*
       (/
        (* (+ alpha beta) (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
        (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
       (-
        (/
         (* (+ alpha beta) (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
         (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
        1.0))))
    2.0)
   (if (or (<= alpha 4.552813519470753e+166)
           (not (<= alpha 4.8907958261153134e+197)))
     (/
      (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
      2.0)
     (/
      (+
       1.0
       (*
        (+ alpha beta)
        (/
         (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
         (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
      2.0))))

C

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 <= 7.158130970526724e+89) {
		tmp = (log(exp(1.0 + pow((((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i)))), 3.0))) / (1.0 + ((((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i)))) * ((((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i)))) - 1.0)))) / 2.0;
	} else if ((alpha <= 4.552813519470753e+166) || !(alpha <= 4.8907958261153134e+197)) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i)))))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 7.15813097052672441e89

   1. Initial program 13.4

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

   3. Applied *-un-lft-identity_binary64 13.4

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

   4. Applied times-frac_binary64 2.9

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

   5. Simplified 2.9

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

   7. Applied flip3-+_binary64 2.9

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

   8. Simplified 2.9

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

   9. Simplified 2.9

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

   11. Applied add-log-exp_binary64 2.9

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

   12. Applied add-log-exp_binary64 2.9

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

   13. Applied sum-log_binary64 2.9

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

   14. Simplified 2.9

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

   if 7.15813097052672441e89 < alpha < 4.55281351947075274e166 or 4.8907958261153134e197 < alpha

   1. Initial program 57.3

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

   2. Taylor expanded around inf 40.1

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

   3. Simplified 40.1

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

   if 4.55281351947075274e166 < alpha < 4.8907958261153134e197

   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. Using strategy rm

   3. Applied *-un-lft-identity_binary64 64.0

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

   4. Applied *-un-lft-identity_binary64 64.0

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

   5. Applied times-frac_binary64 41.8

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

   6. Applied times-frac_binary64 41.7

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

   7. Simplified 41.7

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

   8. Simplified 41.7

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

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.158130970526724 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)}^{3}}\right)}{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - 1\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4.552813519470753 \cdot 10^{+166} \lor \neg \left(\alpha \leq 4.8907958261153134 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))