Average Error: 16.2 → 0.2
Time: 3.3min
Precision: binary64
Cost: 14849
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\ \;\;\;\;\left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) - \left(3 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha \cdot \alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\
\;\;\;\;\left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) - \left(3 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha \cdot \alpha}\right)\right)\\

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

\end{array}
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999999956046)
   (-
    (+ (/ beta alpha) (/ 1.0 alpha))
    (+
     (* 3.0 (/ beta (* alpha alpha)))
     (+ (* (/ beta alpha) (/ beta alpha)) (/ 2.0 (* alpha alpha)))))
   (exp
    (log
     (/
      (-
       (/ beta (+ (+ beta alpha) 2.0))
       (- (/ alpha (+ (+ beta alpha) 2.0)) 1.0))
      2.0)))))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999999956046) {
		tmp = ((beta / alpha) + (1.0 / alpha)) - ((3.0 * (beta / (alpha * alpha))) + (((beta / alpha) * (beta / alpha)) + (2.0 / (alpha * alpha))));
	} else {
		tmp = exp(log(((beta / ((beta + alpha) + 2.0)) - ((alpha / ((beta + alpha) + 2.0)) - 1.0)) / 2.0));
	}
	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

Alternatives

Alternative 1
Error0.2
Cost2689
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\ \;\;\;\;\left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) - \left(3 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha \cdot \alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]
Alternative 2
Error0.2
Cost2049
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]
Alternative 3
Error0.2
Cost1793
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array}\]
Alternative 4
Error0.2
Cost1665
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array}\]
Alternative 5
Error4.5
Cost1025
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 15864504599490.816:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}}{2}\\ \end{array}\]
Alternative 6
Error4.5
Cost897
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 15864504599490.816:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array}\]
Alternative 7
Error27.0
Cost641
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 15864504599490.816:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array}\]
Alternative 8
Error40.0
Cost64
\[1\]
Alternative 9
Error61.6
Cost64
\[0\]

Error

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999999999560463

    1. Initial program 60.4

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

    3. Applied div-sub_binary64_247060.4

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

    4. Applied associate-+l-_binary64_240058.4

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

    5. Simplified58.4

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

    7. Applied frac-2neg_binary64_247658.4

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

    8. Simplified58.4

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

    10. Applied neg-sub0_binary64_246058.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\color{blue}{0 - \alpha}}{-2 - \left(\alpha + \beta\right)} - 1\right)}{2}\]

    11. Applied div-sub_binary64_247058.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\color{blue}{\left(\frac{0}{-2 - \left(\alpha + \beta\right)} - \frac{\alpha}{-2 - \left(\alpha + \beta\right)}\right)} - 1\right)}{2}\]

    12. Applied associate--l-_binary64_240358.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{0}{-2 - \left(\alpha + \beta\right)} - \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + 1\right)\right)}}{2}\]

    13. Simplified58.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{0}{-2 - \left(\alpha + \beta\right)} - \color{blue}{\left(1 + \frac{\alpha}{-2 - \left(\alpha + \beta\right)}\right)}\right)}{2}\]

    14. Taylor expanded around inf 3.2

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

    15. Simplified0.0

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

    16. Simplified0.0

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

    if -0.99999999999560463 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.3

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

    3. Applied div-sub_binary64_24700.3

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

    4. Applied associate-+l-_binary64_24000.3

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

    5. Simplified0.3

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

    7. Applied add-exp-log_binary64_25030.3

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

    8. Applied add-exp-log_binary64_25030.3

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

    9. Applied div-exp_binary64_25160.3

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

    10. Simplified0.3

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

    11. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999956046:\\ \;\;\;\;\left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) - \left(3 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha \cdot \alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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))