Average Error: 3.6 → 1.3
Time: 11.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.0121771842641978 \cdot 10^{159}:\\ \;\;\;\;\frac{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot {\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}^{\left(-2\right)}}{\left(\alpha + \beta\right) + \left(2 \cdot 1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + \left(2 \cdot 1 + 1\right)}\\ \end{array}\]
\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}\;\alpha \le 8.0121771842641978 \cdot 10^{159}:\\
\;\;\;\;\frac{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot {\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}^{\left(-2\right)}}{\left(\alpha + \beta\right) + \left(2 \cdot 1 + 1\right)}\\

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

\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 temp;
	if ((alpha <= 8.012177184264198e+159)) {
		temp = (((((alpha + beta) + (beta * alpha)) + 1.0) * pow(((alpha + beta) + (2.0 * 1.0)), -2.0)) / ((alpha + beta) + ((2.0 * 1.0) + 1.0)));
	} else {
		temp = (((1.0 / (((1.0 / alpha) + (1.0 / beta)) - (1.0 / pow(alpha, 2.0)))) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + ((2.0 * 1.0) + 1.0)));
	}
	return temp;
}

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 alpha < 8.012177184264198e+159

    1. Initial program 1.2

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

    3. Applied associate-+l+1.2

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

    5. Applied *-un-lft-identity1.2

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

    6. Applied div-inv1.2

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

    7. Applied times-frac1.6

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

    8. Simplified1.6

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

    10. Applied pow11.6

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

    11. Applied pow11.6

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

    12. Applied pow-flip1.6

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

    13. Applied pow-div1.6

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

    14. Simplified1.6

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

    if 8.012177184264198e+159 < alpha

    1. Initial program 15.8

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

    3. Applied associate-+l+15.8

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

    5. Applied clear-num15.8

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

    6. Taylor expanded around inf 0.1

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.0121771842641978 \cdot 10^{159}:\\ \;\;\;\;\frac{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot {\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}^{\left(-2\right)}}{\left(\alpha + \beta\right) + \left(2 \cdot 1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + \left(2 \cdot 1 + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(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)))