Average Error: 3.6 → 1.1
Time: 35.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.0121771842641978 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\sqrt{1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\left(\mathsf{fma}\left(\alpha, \beta, \beta\right) + \alpha\right) + 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1}}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}\\ \end{array}\]
\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{\frac{\frac{\frac{\sqrt{1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\left(\mathsf{fma}\left(\alpha, \beta, \beta\right) + \alpha\right) + 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1}}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\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 = ((((sqrt(1.0) / fma(1.0, 2.0, (alpha + beta))) / 1.0) / (((alpha + beta) + fma(2.0, 1.0, 1.0)) / ((fma(alpha, beta, beta) + alpha) + 1.0))) / ((alpha + beta) + (2.0 * 1.0)));
	} else {
		temp = (((sqrt(1.0) / fma(1.0, 2.0, (alpha + beta))) / 1.0) / (2.0 + ((beta / alpha) + (alpha / beta))));
	}
	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. Simplified1.2

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

    6. Applied clear-num1.2

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

    7. Simplified1.2

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

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

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

    10. Applied div-inv1.3

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

    11. Applied add-sqr-sqrt1.3

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

    12. Applied times-frac1.2

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

    13. Applied times-frac1.2

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

    14. Applied associate-/l*1.2

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

    15. Simplified1.2

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

    17. Applied associate-/r/1.2

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

    18. Applied associate-/r*1.2

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

    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. Simplified15.8

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

    6. Applied clear-num15.8

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

    7. Simplified15.8

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

    9. Applied *-un-lft-identity15.8

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

    10. Applied div-inv15.8

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

    11. Applied add-sqr-sqrt15.8

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

    12. Applied times-frac15.8

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

    13. Applied times-frac15.8

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

    14. Applied associate-/l*15.8

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

    15. Simplified15.8

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

    16. Taylor expanded around inf 0.1

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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)))