Average Error: 16.1 → 0.1
Time: 11.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999735213011:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\frac{1 - \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}{-1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}{2}\\ \end{array}\]
\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 -0.9999999735213011:\\
\;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\frac{1 - \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}{-1 - \frac{\alpha}{\alpha + \left(\beta + 2\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)) -0.9999999735213011)
   (/
    (+
     (-
      (+ (/ 2.0 alpha) (* (/ beta alpha) (- 2.0 (/ 6.0 alpha))))
      (/ 4.0 (* alpha alpha)))
     (* (* (/ beta alpha) (/ beta alpha)) -2.0))
    2.0)
   (/
    (-
     (/ beta (+ (+ beta alpha) 2.0))
     (/
      (/
       (-
        1.0
        (*
         (* (/ alpha (+ alpha (+ beta 2.0))) (/ alpha (+ alpha (+ beta 2.0))))
         (*
          (/ alpha (+ alpha (+ beta 2.0)))
          (/ alpha (+ alpha (+ beta 2.0))))))
       (+
        1.0
        (* (/ alpha (+ alpha (+ beta 2.0))) (/ alpha (+ alpha (+ beta 2.0))))))
      (- -1.0 (/ alpha (+ alpha (+ beta 2.0))))))
    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)) <= -0.9999999735213011) {
		tmp = ((((2.0 / alpha) + ((beta / alpha) * (2.0 - (6.0 / alpha)))) - (4.0 / (alpha * alpha))) + (((beta / alpha) * (beta / alpha)) * -2.0)) / 2.0;
	} else {
		tmp = ((beta / ((beta + alpha) + 2.0)) - (((1.0 - (((alpha / (alpha + (beta + 2.0))) * (alpha / (alpha + (beta + 2.0)))) * ((alpha / (alpha + (beta + 2.0))) * (alpha / (alpha + (beta + 2.0)))))) / (1.0 + ((alpha / (alpha + (beta + 2.0))) * (alpha / (alpha + (beta + 2.0)))))) / (-1.0 - (alpha / (alpha + (beta + 2.0)))))) / 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

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.9999999735213011

    1. Initial program 59.8

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

    2. Taylor expanded around inf 3.1

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

    3. Simplified0.0

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

    if -0.9999999735213011 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.1

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

    3. Applied div-sub_binary64_11060.1

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

    4. Applied associate-+l-_binary64_10360.1

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

    5. Simplified0.1

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

    7. Applied flip-+_binary64_10750.1

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

    8. Simplified0.1

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

    10. Applied flip--_binary64_10760.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999735213011:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\frac{1 - \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}{-1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))