Average Error: 52.9 → 36.3
Time: 7.4s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.02655186405084379 \cdot 10^{207}:\\ \;\;\;\;\frac{\sqrt{\frac{i}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}{\alpha + \left(\beta + \left(i \cdot 2 + \sqrt{1}\right)\right)} \cdot \frac{\sqrt{\frac{i}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}{\alpha + \left(\beta + \left(i \cdot 2 - \sqrt{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\beta \le 2.02655186405084379 \cdot 10^{207}:\\
\;\;\;\;\frac{\sqrt{\frac{i}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}{\alpha + \left(\beta + \left(i \cdot 2 + \sqrt{1}\right)\right)} \cdot \frac{\sqrt{\frac{i}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}{\alpha + \left(\beta + \left(i \cdot 2 - \sqrt{1}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 ((beta <= 2.0265518640508438e+207)) {
		VAR = ((double) (((double) (((double) sqrt(((double) (((double) (i / ((double) (((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))) / ((double) (i + ((double) (beta + alpha)))))))) * ((double) (((double) (((double) (beta * alpha)) + ((double) (i * ((double) (i + ((double) (beta + alpha)))))))) / ((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))))))))) / ((double) (alpha + ((double) (beta + ((double) (((double) (i * 2.0)) + ((double) sqrt(1.0)))))))))) * ((double) (((double) sqrt(((double) (((double) (i / ((double) (((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))) / ((double) (i + ((double) (beta + alpha)))))))) * ((double) (((double) (((double) (beta * alpha)) + ((double) (i * ((double) (i + ((double) (beta + alpha)))))))) / ((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))))))))) / ((double) (alpha + ((double) (beta + ((double) (((double) (i * 2.0)) - ((double) sqrt(1.0))))))))))));
	} else {
		VAR = 0.0;
	}
	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 beta < 2.02655186405084379e207

    1. Initial program 51.6

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

    3. Applied add-sqr-sqrt51.6

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

    4. Applied difference-of-squares51.7

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

    5. Applied add-sqr-sqrt51.6

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

    6. Applied times-frac51.7

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

    7. Simplified51.6

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

    8. Simplified35.3

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

    if 2.02655186405084379e207 < beta

    1. Initial program 62.6

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

    2. Taylor expanded around inf 44.5

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification36.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.02655186405084379 \cdot 10^{207}:\\ \;\;\;\;\frac{\sqrt{\frac{i}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}{\alpha + \left(\beta + \left(i \cdot 2 + \sqrt{1}\right)\right)} \cdot \frac{\sqrt{\frac{i}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}{\alpha + \left(\beta + \left(i \cdot 2 - \sqrt{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))