Average Error: 3.7 → 1.3
Time: 10.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.0706985136143749 \cdot 10^{+161}:\\ \;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\sqrt{\frac{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]
\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 \leq 1.0706985136143749 \cdot 10^{+161}:\\
\;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\sqrt{\frac{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\

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

\end{array}
(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ 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)))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.0706985136143749e+161)
   (*
    (/
     (sqrt
      (/
       (+ alpha (+ beta (+ (* beta alpha) 1.0)))
       (+ alpha (+ beta (* 1.0 2.0)))))
     (sqrt (+ alpha (+ beta (* 1.0 2.0)))))
    (/
     (sqrt
      (/
       (+ alpha (+ beta (+ (* beta alpha) 1.0)))
       (+ alpha (+ beta (* 1.0 2.0)))))
     (*
      (sqrt (+ alpha (+ beta (* 1.0 2.0))))
      (+ alpha (+ beta (+ 1.0 (* 1.0 2.0)))))))
   (/
    1.0
    (*
     (+ alpha (+ beta (* 1.0 2.0)))
     (+ 2.0 (+ (/ beta alpha) (/ alpha beta)))))))
double code(double alpha, double beta) {
	return (((((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0)));
}
↓
double code(double alpha, double beta) {
	double tmp;
	if ((beta <= 1.0706985136143749e+161)) {
		tmp = ((double) ((((double) sqrt((((double) (alpha + ((double) (beta + ((double) (((double) (beta * alpha)) + 1.0)))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0))))))))) / ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0))))))))) * (((double) sqrt((((double) (alpha + ((double) (beta + ((double) (((double) (beta * alpha)) + 1.0)))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0))))))))) / ((double) (((double) sqrt(((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))))) * ((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (1.0 * 2.0)))))))))))));
	} else {
		tmp = (1.0 / ((double) (((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))) * ((double) (2.0 + ((double) ((beta / alpha) + (alpha / beta))))))));
	}
	return tmp;
}

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 < 1.07069851361437491e161

    1. Initial program Error: 1.3 bits

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

    3. Applied *-un-lft-identityError: 1.3 bits

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

    4. Applied add-sqr-sqrtError: 1.9 bits

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

    5. Applied add-sqr-sqrtError: 1.5 bits

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

    6. Applied times-fracError: 1.5 bits

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

    7. Applied times-fracError: 1.5 bits

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

    8. SimplifiedError: 1.5 bits

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

    9. SimplifiedError: 1.5 bits

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

    if 1.07069851361437491e161 < beta

    1. Initial program Error: 16.3 bits

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

    3. Applied clear-numError: 16.4 bits

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

    4. SimplifiedError: 16.4 bits

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

    5. Taylor expanded around inf Error: 0.5 bits

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

  4. Final simplificationError: 1.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.0706985136143749 \cdot 10^{+161}:\\ \;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\sqrt{\frac{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ 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)))