Octave 3.8, jcobi/1

Average Error: 16.3 → 0.1

Time: 11.0s

Precision: binary64

\[\alpha > -1 \land \beta > -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999997918492348:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) + \left(\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2 + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \log \left({\left(e^{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}\right)}^{\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} + -1\right)}\right)}}{2}\\ \end{array}\]

FPCore

(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)) -0.9999997918492348)
   (/
    (+
     (+ (/ 2.0 alpha) (* (/ beta alpha) (- 2.0 (/ 6.0 alpha))))
     (+ (* (* (/ beta alpha) (/ beta alpha)) -2.0) (/ -4.0 (* alpha alpha))))
    2.0)
   (/
    (/
     (+ 1.0 (pow (/ (- beta alpha) (+ alpha (+ beta 2.0))) 3.0))
     (+
      1.0
      (log
       (pow
        (exp (/ (- beta alpha) (+ alpha (+ beta 2.0))))
        (+ (/ (- beta alpha) (+ alpha (+ beta 2.0))) -1.0)))))
    2.0)))

C

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)) <= -0.9999997918492348) {
		tmp = (((2.0 / alpha) + ((beta / alpha) * (2.0 - (6.0 / alpha)))) + ((((beta / alpha) * (beta / alpha)) * -2.0) + (-4.0 / (alpha * alpha)))) / 2.0;
	} else {
		tmp = ((1.0 + pow(((beta - alpha) / (alpha + (beta + 2.0))), 3.0)) / (1.0 + log(pow(exp((beta - alpha) / (alpha + (beta + 2.0))), (((beta - alpha) / (alpha + (beta + 2.0))) + -1.0))))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999979184923482

   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. Simplified 0.1

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

   if -0.99999979184923482 < (/.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 flip3-+_binary64_2809 0.1

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

   4. Simplified 0.1

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

   5. Simplified 0.1

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

   7. Applied add-log-exp_binary64_2845 0.1

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

   8. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999997918492348:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) + \left(\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2 + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \log \left({\left(e^{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}\right)}^{\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} + -1\right)}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021016 
(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))