Average Error: 53.8 → 10.2
Time: 7.4s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 3.3078638610239012 \cdot 10^{+100}:\\ \;\;\;\;\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \left(\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + \left(i \cdot 2 + \sqrt{1}\right)\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + \left(i \cdot 2 - \sqrt{1}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \frac{i \cdot 0.5 + \left(0.125 \cdot \frac{1}{i} - \sqrt{1} \cdot 0.25\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - \sqrt{1}}\\ \end{array}\]
\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 \leq 3.3078638610239012 \cdot 10^{+100}:\\
\;\;\;\;\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \left(\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + \left(i \cdot 2 + \sqrt{1}\right)\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + \left(i \cdot 2 - \sqrt{1}\right)\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \frac{i \cdot 0.5 + \left(0.125 \cdot \frac{1}{i} - \sqrt{1} \cdot 0.25\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - \sqrt{1}}\\

\end{array}
double code(double alpha, double beta, double i) {
	return ((((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 <= 3.3078638610239012e+100)) {
		VAR = ((double) (((double) ((i / ((double) (alpha + ((double) (beta + ((double) (i * 2.0))))))) * (((double) (i + ((double) (alpha + beta)))) / ((double) (alpha + ((double) (beta + ((double) (i * 2.0))))))))) * ((double) ((((double) sqrt(((double) (((double) (i * ((double) (i + ((double) (alpha + beta)))))) + ((double) (alpha * beta)))))) / ((double) (alpha + ((double) (beta + ((double) (((double) (i * 2.0)) + ((double) sqrt(1.0))))))))) * (((double) sqrt(((double) (((double) (i * ((double) (i + ((double) (alpha + beta)))))) + ((double) (alpha * beta)))))) / ((double) (alpha + ((double) (beta + ((double) (((double) (i * 2.0)) - ((double) sqrt(1.0)))))))))))));
	} else {
		VAR = ((double) (((double) ((i / ((double) (alpha + ((double) (beta + ((double) (i * 2.0))))))) * (((double) (i + ((double) (alpha + beta)))) / ((double) (alpha + ((double) (beta + ((double) (i * 2.0))))))))) * (((double) (((double) (i * 0.5)) + ((double) (((double) (0.125 * (1.0 / i))) - ((double) (((double) sqrt(1.0)) * 0.25)))))) / ((double) (((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))) - ((double) sqrt(1.0)))))));
	}
	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 < 3.3078638610239012e100

    1. Initial program 32.8

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

    2. Simplified18.6

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

    4. Applied *-un-lft-identity18.6

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

    5. Applied times-frac13.5

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

    6. Applied associate-*r*13.5

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

    7. Simplified13.4

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

    9. Applied add-sqr-sqrt13.4

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

    10. Applied difference-of-squares13.4

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

    11. Applied add-sqr-sqrt13.4

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

    12. Applied times-frac8.3

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

    13. Simplified8.3

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

    14. Simplified8.3

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

    if 3.3078638610239012e100 < 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. Simplified63.2

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

    4. Applied *-un-lft-identity63.2

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

    5. Applied times-frac52.2

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

    6. Applied associate-*r*52.2

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

    7. Simplified52.1

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

    9. Applied add-sqr-sqrt52.1

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

    10. Applied difference-of-squares52.1

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

    11. Applied associate-/r*51.2

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

    12. Simplified51.2

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

    13. Taylor expanded around inf 11.1

      \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \frac{\color{blue}{\left(0.5 \cdot i + 0.125 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{i}\right) - 0.25 \cdot \sqrt{1}}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - \sqrt{1}}\]

    14. Simplified11.1

      \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \frac{\color{blue}{i \cdot 0.5 + \left(0.125 \cdot \frac{1}{i} - \sqrt{1} \cdot 0.25\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - \sqrt{1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.3078638610239012 \cdot 10^{+100}:\\ \;\;\;\;\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \left(\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + \left(i \cdot 2 + \sqrt{1}\right)\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + \left(i \cdot 2 - \sqrt{1}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right) \cdot \frac{i \cdot 0.5 + \left(0.125 \cdot \frac{1}{i} - \sqrt{1} \cdot 0.25\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))