Average Error: 3.7 → 2.4
Time: 5.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.1295199514491574 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\frac{1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}} \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.1295199514491574 \cdot 10^{182}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\frac{1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}} \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double code(double alpha, double beta) {
	return ((((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0));
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((beta <= 1.1295199514491574e+182)) {
		VAR = (((sqrt((((alpha + beta) + (beta * alpha)) + 1.0)) / sqrt(((alpha + beta) + (2.0 * 1.0)))) / sqrt(((alpha + beta) + (2.0 * 1.0)))) / (((1.0 / sqrt(sqrt((((alpha + beta) + (beta * alpha)) + 1.0)))) * ((((alpha + beta) + (2.0 * 1.0)) + 1.0) / sqrt(sqrt((((alpha + beta) + (beta * alpha)) + 1.0))))) * ((alpha + beta) + (2.0 * 1.0))));
	} else {
		VAR = (0.0 / (((alpha + beta) + (2.0 * 1.0)) + 1.0));
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.1295199514491574e+182

    1. Initial program 1.6

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

    3. Applied add-sqr-sqrt2.2

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

    4. Applied add-sqr-sqrt2.5

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

    5. Applied add-sqr-sqrt2.5

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

    6. Applied times-frac2.5

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

    7. Applied times-frac2.2

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

    8. Applied associate-/l*2.2

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

    9. Simplified1.7

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

    11. Applied add-sqr-sqrt1.7

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

    12. Applied sqrt-prod1.8

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

    13. Applied *-un-lft-identity1.8

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

    14. Applied times-frac1.8

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

    if 1.1295199514491574e+182 < beta

    1. Initial program 17.2

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

    3. Applied *-un-lft-identity17.2

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

    4. Applied div-inv17.2

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

    5. Applied times-frac17.5

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

    6. Simplified17.5

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

    7. Taylor expanded around inf 6.6

      \[\leadsto \frac{\color{blue}{0}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.1295199514491574 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\frac{1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}} \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))