Average Error: 16.3 → 0.1
Time: 8.3s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999994945147518:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{\beta}{\alpha} \cdot \left(\left(2 - \frac{6}{\alpha}\right) + \frac{\beta}{\alpha} \cdot -2\right) + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{2}\\ \end{array}\]
\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 -0.9999994945147518:\\
\;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{\beta}{\alpha} \cdot \left(\left(2 - \frac{6}{\alpha}\right) + \frac{\beta}{\alpha} \cdot -2\right) + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{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)) -0.9999994945147518)
   (/
    (+
     (/ 2.0 alpha)
     (+
      (* (/ beta alpha) (+ (- 2.0 (/ 6.0 alpha)) (* (/ beta alpha) -2.0)))
      (/ -4.0 (* alpha alpha))))
    2.0)
   (/ (+ 1.0 (/ 1.0 (/ (+ (+ beta alpha) 2.0) (- beta alpha)))) 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)) <= -0.9999994945147518) {
		tmp = ((2.0 / alpha) + (((beta / alpha) * ((2.0 - (6.0 / alpha)) + ((beta / alpha) * -2.0))) + (-4.0 / (alpha * alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((beta + alpha) + 2.0) / (beta - alpha)))) / 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)) < -0.999999494514751786

    1. Initial program 59.4

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

    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. Simplified0.1

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

    5. Applied associate-+l+_binary64_17160.1

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

    6. Simplified0.1

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

    if -0.999999494514751786 < (/.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 clear-num_binary64_17820.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999994945147518:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{\beta}{\alpha} \cdot \left(\left(2 - \frac{6}{\alpha}\right) + \frac{\beta}{\alpha} \cdot -2\right) + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(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))