Average Error: 53.6 → 15.7
Time: 6.0s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 2.66652702391726766 \cdot 10^{83}:\\ \;\;\;\;\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot {e}^{\left(\log \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) - \left(\log \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1\right) + 2 \cdot \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}
↓
\begin{array}{l}
\mathbf{if}\;i \le 2.66652702391726766 \cdot 10^{83}:\\
\;\;\;\;\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot {e}^{\left(\log \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) - \left(\log \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1\right) + 2 \cdot \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;0.0625\\

\end{array}
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (i * ((double) (((double) (alpha + beta)) + i)))) * ((double) (((double) (beta * alpha)) + ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))))) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) - 1.0))));
}
↓
double code(double alpha, double beta, double i) {
	double VAR;
	if ((i <= 2.6665270239172677e+83)) {
		VAR = ((double) (((double) (i * ((double) (i + ((double) (alpha + beta)))))) * ((double) pow(((double) M_E), ((double) (((double) log(((double) (((double) (i * ((double) (i + ((double) (alpha + beta)))))) + ((double) (alpha * beta)))))) - ((double) (((double) log(((double) (((double) (((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))) * ((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))))) - 1.0)))) + ((double) (2.0 * ((double) log(((double) (alpha + ((double) (beta + ((double) (i * 2.0))))))))))))))))));
	} else {
		VAR = 0.0625;
	}
	return VAR;
}

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 2 regimes

  2. if i < 2.66652702391726766e83

    1. Initial program 26.3

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

    2. Simplified22.7

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

    4. Applied add-exp-log25.1

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

    5. Applied add-exp-log25.3

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

    6. Applied prod-exp25.6

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

    7. Applied add-exp-log25.7

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

    8. Applied prod-exp25.6

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

    9. Applied add-exp-log25.5

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

    10. Applied div-exp19.2

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

    11. Simplified19.2

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

    13. Applied *-un-lft-identity19.2

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

    14. Applied exp-prod19.0

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

    15. Simplified19.0

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

    if 2.66652702391726766e83 < i

    1. Initial program 64.0

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

    2. Simplified63.5

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

    4. Applied add-exp-log63.5

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

    5. Applied add-exp-log63.5

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

    6. Applied prod-exp63.5

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

    7. Applied add-exp-log63.5

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

    8. Applied prod-exp63.5

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

    9. Applied add-exp-log63.5

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

    10. Applied div-exp52.3

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

    11. Simplified52.3

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

    12. Taylor expanded around inf 52.8

      \[\leadsto \color{blue}{{i}^{2} \cdot e^{\left(\log 1 + 2 \cdot \log \left(\frac{1}{i}\right)\right) - \left(\log 4 + 2 \cdot \log 2\right)}}\]

    13. Simplified14.5

      \[\leadsto \color{blue}{i \cdot \frac{\frac{1}{i}}{4 \cdot \left(2 \cdot 2\right)}}\]

    14. Taylor expanded around 0 14.4

      \[\leadsto \color{blue}{0.0625}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.66652702391726766 \cdot 10^{83}:\\ \;\;\;\;\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot {e}^{\left(\log \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) - \left(\log \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1\right) + 2 \cdot \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))