Average Error: 23.4 → 11.7
Time: 40.1s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\frac{\frac{\log \left(e^{1 + {\left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3}}\right)}{1 + \left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} - {\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}}{\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + \left(1 - \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}}{2}\]
\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}
↓
\frac{\frac{\frac{\log \left(e^{1 + {\left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3}}\right)}{1 + \left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} - {\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}}{\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + \left(1 - \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}}{2}
(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
 (/
  (/
   (/
    (log
     (exp
      (+
       1.0
       (pow
        (pow
         (/
          (+ alpha beta)
          (/
           (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))
           (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))))
         3.0)
        3.0))))
    (+
     1.0
     (-
      (pow
       (/
        (+ alpha beta)
        (/
         (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))
         (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))))
       6.0)
      (pow
       (/
        (+ alpha beta)
        (/
         (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))
         (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))))
       3.0))))
   (+
    (*
     (/
      (+ alpha beta)
      (/
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))
       (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))))
     (/
      (+ alpha beta)
      (/
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))
       (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))))
    (-
     1.0
     (/
      (+ alpha beta)
      (/
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))
       (/ (- beta alpha) (+ (+ 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) {
	return ((log(exp(1.0 + pow(pow(((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / ((beta - alpha) / ((alpha + beta) + (2.0 * i))))), 3.0), 3.0))) / (1.0 + (pow(((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / ((beta - alpha) / ((alpha + beta) + (2.0 * i))))), 6.0) - pow(((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / ((beta - alpha) / ((alpha + beta) + (2.0 * i))))), 3.0)))) / ((((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / ((beta - alpha) / ((alpha + beta) + (2.0 * i))))) * ((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))) + (1.0 - ((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / ((beta - alpha) / ((alpha + beta) + (2.0 * i)))))))) / 2.0;
}

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. Initial program 23.4

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

  3. Applied *-un-lft-identity_binary6423.4

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

  4. Applied times-frac_binary6411.7

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

  5. Applied associate-/l*_binary6411.7

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

  6. Simplified11.7

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

  8. Applied flip3-+_binary6411.7

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

  9. Simplified11.7

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

  10. Simplified11.7

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

  12. Applied flip3-+_binary6411.7

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

  13. Simplified11.7

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

  14. Simplified11.7

    \[\leadsto \frac{\frac{\frac{1 + {\left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3}}{\color{blue}{1 + \left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} - {\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}}}{\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + \left(1 - \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}}{2}\]
  15. Using strategy rm

  16. Applied add-log-exp_binary6411.7

    \[\leadsto \frac{\frac{\frac{\color{blue}{\log \left(e^{1 + {\left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3}}\right)}}{1 + \left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} - {\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}}{\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + \left(1 - \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}}{2}\]

  17. Final simplification11.7

    \[\leadsto \frac{\frac{\frac{\log \left(e^{1 + {\left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}^{3}}\right)}{1 + \left({\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{6} - {\left(\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}^{3}\right)}}{\frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + \left(1 - \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}}{2}\]

Reproduce

herbie shell --seed 2020277 
(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))