Average Error: 3.7 → 2.3
Time: 48.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.156274647039849 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + \left(1 + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \sqrt{0.5} + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 7.156274647039849 \cdot 10^{+214}:\\
\;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + \left(1 + 1 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \sqrt{0.5} + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\

\end{array}
(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7.156274647039849e+214)
   (/
    (/
     (/
      (+ (+ (+ alpha beta) (* alpha beta)) 1.0)
      (+ (+ alpha beta) (* 1.0 2.0)))
     (+ (+ alpha beta) (* 1.0 2.0)))
    (+ (+ alpha beta) (+ 1.0 (* 1.0 2.0))))
   (/
    (/
     (/
      (-
       (+
        (* beta (sqrt 0.5))
        (+ (* 0.75 (* alpha (sqrt 0.5))) (* 1.0 (sqrt 0.5))))
       (* 0.125 (/ beta (sqrt 0.5))))
      (sqrt (+ (+ alpha beta) (* 1.0 2.0))))
     (+ (+ alpha beta) (* 1.0 2.0)))
    (+ 1.0 (+ (+ alpha beta) (* 1.0 2.0))))))
double code(double alpha, double beta) {
	return (((((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0)));
}
↓
double code(double alpha, double beta) {
	double tmp;
	if ((alpha <= 7.156274647039849e+214)) {
		tmp = (((((double) (((double) (((double) (alpha + beta)) + ((double) (alpha * beta)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0))))) / ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0))))) / ((double) (((double) (alpha + beta)) + ((double) (1.0 + ((double) (1.0 * 2.0)))))));
	} else {
		tmp = (((((double) (((double) (((double) (beta * ((double) sqrt(0.5)))) + ((double) (((double) (0.75 * ((double) (alpha * ((double) sqrt(0.5)))))) + ((double) (1.0 * ((double) sqrt(0.5)))))))) - ((double) (0.125 * (beta / ((double) sqrt(0.5))))))) / ((double) sqrt(((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0))))))) / ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0))))) / ((double) (1.0 + ((double) (((double) (alpha + beta)) + ((double) (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

Derivation

  1. Split input into 2 regimes

  2. if alpha < 7.15627464703984901e214

    1. Initial program 2.1

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

    3. Applied associate-+l+_binary642.1

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

    4. Simplified2.1

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

    if 7.15627464703984901e214 < alpha

    1. Initial program 18.0

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

    3. Applied add-sqr-sqrt_binary6418.0

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

    4. Applied associate-/r*_binary6418.0

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

    5. Simplified18.0

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

    6. Taylor expanded around 0 4.8

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \beta + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.156274647039849 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + \left(1 + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \sqrt{0.5} + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))