Average Error: 16.4 → 0.5
Time: 4.9min
Precision: binary64
Cost: 14849
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999992820278:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\beta + 2} - \log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1}\right)}{2}\\ \end{array}\]
\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 -0.9999999992820278:\\
\;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\beta \cdot \frac{1}{\beta + 2} - \log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1}\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 (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999992820278)
   (/
    (-
     (+ (* 2.0 (/ beta alpha)) (/ 2.0 alpha))
     (+
      (* 6.0 (/ beta (* alpha alpha)))
      (+ (/ 4.0 (* alpha alpha)) (* 2.0 (* (/ beta alpha) (/ beta alpha))))))
    2.0)
   (/
    (-
     (* beta (/ 1.0 (+ beta 2.0)))
     (log (exp (- (/ 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)) <= -0.9999999992820278) {
		tmp = (((2.0 * (beta / alpha)) + (2.0 / alpha)) - ((6.0 * (beta / (alpha * alpha))) + ((4.0 / (alpha * alpha)) + (2.0 * ((beta / alpha) * (beta / alpha)))))) / 2.0;
	} else {
		tmp = ((beta * (1.0 / (beta + 2.0))) - log(exp((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
Cost3073
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999992820278:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array}\]
Alternative 2
Error0.2
Cost1793
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999992820278:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array}\]
Alternative 3
Error0.2
Cost1793
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999992820278:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array}\]
Alternative 4
Error0.2
Cost1665
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999992820278:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array}\]
Alternative 5
Error4.9
Cost1153
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.280321599555968 \cdot 10^{+31}:\\ \;\;\;\;\frac{1 + \beta \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array}\]
Alternative 6
Error4.9
Cost897
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.1580978541201579 \cdot 10^{+32}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array}\]
Alternative 7
Error17.7
Cost576
\[\frac{1 + \frac{\beta}{\beta + 2}}{2}\]
Alternative 8
Error18.6
Cost385
\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.9929106022751548:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 9
Error40.7
Cost64
\[1\]
Alternative 10
Error61.6
Cost64
\[0\]

Error

Time

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999999282027763

    1. Initial program 60.0

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

    2. Taylor expanded around inf 2.5

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - \left(6 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + 2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)}}{2}\]

    3. Simplified0.0

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)}}{2}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)}{2}}\]

    if -0.999999999282027763 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.2

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

    3. Applied div-sub_binary64_21290.2

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

    4. Applied associate-+l-_binary64_20590.2

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

    5. Simplified0.2

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

    7. Applied div-inv_binary64_21210.2

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

    9. Applied add-log-exp_binary64_21630.2

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

    10. Applied add-log-exp_binary64_21630.2

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

    11. Applied diff-log_binary64_22160.2

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

    12. Simplified0.2

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

    13. Taylor expanded around 0 0.7

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

    14. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999992820278:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\beta + 2} - \log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(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))