Average Error: 16.0 → 0.2
Time: 1.3min
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999997426335:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{4}{\alpha \cdot \alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}\right)}{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{-1 - \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}}}{2}\\ \end{array}\]
\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 -0.9999999997426335:\\
\;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{4}{\alpha \cdot \alpha}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}\right)}{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{-1 - \frac{\beta - \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)) -0.9999999997426335)
   (/
    (-
     (+ (* 2.0 (/ beta alpha)) (/ 2.0 alpha))
     (+
      (* 6.0 (/ beta (* alpha alpha)))
      (+ (* 2.0 (* (/ beta alpha) (/ beta alpha))) (/ 4.0 (* alpha alpha)))))
    2.0)
   (/
    (/
     (log (exp (+ 1.0 (pow (/ (- beta alpha) (+ alpha (+ beta 2.0))) 3.0))))
     (+
      1.0
      (*
       (/ (- beta alpha) (+ alpha (+ beta 2.0)))
       (/
        (-
         1.0
         (*
          (/ (- beta alpha) (+ alpha (+ beta 2.0)))
          (/ (- beta alpha) (+ alpha (+ beta 2.0)))))
        (- -1.0 (/ (- beta 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)) <= -0.9999999997426335) {
		tmp = (((2.0 * (beta / alpha)) + (2.0 / alpha)) - ((6.0 * (beta / (alpha * alpha))) + ((2.0 * ((beta / alpha) * (beta / alpha))) + (4.0 / (alpha * alpha))))) / 2.0;
	} else {
		tmp = (log(exp(1.0 + pow(((beta - alpha) / (alpha + (beta + 2.0))), 3.0))) / (1.0 + (((beta - alpha) / (alpha + (beta + 2.0))) * ((1.0 - (((beta - alpha) / (alpha + (beta + 2.0))) * ((beta - alpha) / (alpha + (beta + 2.0))))) / (-1.0 - ((beta - 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)) < -0.999999999742633539

    1. Initial program 60.0

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

    3. Applied add-cbrt-cube_binary64_250160.1

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

    4. Simplified60.1

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

    5. Taylor expanded around inf 3.1

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

    6. Simplified0.0

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

    if -0.999999999742633539 < (/.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 flip3-+_binary64_24680.2

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

    4. Simplified0.2

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

    5. Simplified0.2

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

    7. Applied add-log-exp_binary64_25040.2

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

    8. Applied add-log-exp_binary64_25040.2

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

    9. Applied sum-log_binary64_25560.2

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

    10. Simplified0.2

      \[\leadsto \frac{\frac{\log \color{blue}{\left(e^{1 + {\left(\frac{\beta - \alpha}{\alpha + \left(2 + \beta\right)}\right)}^{3}}\right)}}{1 + \frac{\beta - \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(-1 + \frac{\beta - \alpha}{\alpha + \left(2 + \beta\right)}\right)}}{2}\]
    11. Using strategy rm

    12. Applied flip-+_binary64_24390.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999997426335:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{4}{\alpha \cdot \alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}\right)}{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{-1 - \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))