Average Error: 15.4 → 6.1
Time: 3.1s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 825090542879979.5:\\ \;\;\;\;\frac{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta}} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 9.366035363736922 \cdot 10^{+121} \lor \neg \left(\alpha \leq 1.1231273814540445 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\alpha + \left(\beta + 2\right)}\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 825090542879979.5:\\
\;\;\;\;\frac{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta}} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\

\mathbf{elif}\;\alpha \leq 9.366035363736922 \cdot 10^{+121} \lor \neg \left(\alpha \leq 1.1231273814540445 \cdot 10^{+140}\right):\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

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

\end{array}
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 825090542879979.5)
   (/
    (+
     (/ 1.0 (/ (+ alpha (+ beta 2.0)) beta))
     (- 1.0 (/ alpha (+ beta (+ alpha 2.0)))))
    2.0)
   (if (or (<= alpha 9.366035363736922e+121)
           (not (<= alpha 1.1231273814540445e+140)))
     (/
      (+
       (/ beta (+ 2.0 (+ alpha beta)))
       (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha))))
      2.0)
     (/
      (+
       (exp (log (/ beta (+ alpha (+ beta 2.0)))))
       (- 1.0 (/ alpha (+ beta (+ alpha 2.0)))))
      2.0))))
double code(double alpha, double beta) {
	return (((double) ((((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0))) + 1.0)) / 2.0);
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 825090542879979.5)) {
		VAR = (((double) ((1.0 / (((double) (alpha + ((double) (beta + 2.0)))) / beta)) + ((double) (1.0 - (alpha / ((double) (beta + ((double) (alpha + 2.0))))))))) / 2.0);
	} else {
		double VAR_1;
		if (((alpha <= 9.366035363736922e+121) || !(alpha <= 1.1231273814540445e+140))) {
			VAR_1 = (((double) ((beta / ((double) (2.0 + ((double) (alpha + beta))))) + ((double) (((double) ((2.0 / alpha) + (8.0 / ((double) pow(alpha, 3.0))))) - (4.0 / ((double) (alpha * alpha))))))) / 2.0);
		} else {
			VAR_1 = (((double) (((double) exp(((double) log((beta / ((double) (alpha + ((double) (beta + 2.0))))))))) + ((double) (1.0 - (alpha / ((double) (beta + ((double) (alpha + 2.0))))))))) / 2.0);
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if alpha < 825090542879979.5

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    5. Simplified0.3

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

    7. Applied clear-num0.3

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

    8. Simplified0.3

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

    if 825090542879979.5 < alpha < 9.36603536373692181e121 or 1.12312738145404454e140 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-49.0

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

    5. Simplified49.0

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

    6. Taylor expanded around inf 17.6

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    7. Simplified17.6

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]

    if 9.36603536373692181e121 < alpha < 1.12312738145404454e140

    1. Initial program 49.6

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

    3. Applied div-sub49.6

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

    4. Applied associate-+l-48.0

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

    5. Simplified48.0

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

    7. Applied add-exp-log49.1

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

    8. Applied add-exp-log49.0

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

    9. Applied div-exp49.0

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

    10. Simplified48.6

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 825090542879979.5:\\ \;\;\;\;\frac{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta}} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 9.366035363736922 \cdot 10^{+121} \lor \neg \left(\alpha \leq 1.1231273814540445 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\alpha + \left(\beta + 2\right)}\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))