Average Error: 15.9 → 6.0
Time: 5.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.9257281358558516 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 2.9257281358558516 \cdot 10^{+23}:\\
\;\;\;\;\frac{1 + \frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\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 (<= alpha 2.9257281358558516e+23)
   (/
    (+
     1.0
     (*
      (/ 1.0 (sqrt (+ (+ alpha beta) 2.0)))
      (/ (- beta alpha) (sqrt (+ (+ alpha beta) 2.0)))))
    2.0)
   (/
    (-
     (/ (* (cbrt beta) (cbrt beta)) (/ (+ (+ alpha beta) 2.0) (cbrt beta)))
     (- (/ 4.0 (* alpha alpha)) (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0)))))
    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 (alpha <= 2.9257281358558516e+23) {
		tmp = (1.0 + ((1.0 / sqrt((alpha + beta) + 2.0)) * ((beta - alpha) / sqrt((alpha + beta) + 2.0)))) / 2.0;
	} else {
		tmp = (((cbrt(beta) * cbrt(beta)) / (((alpha + beta) + 2.0) / cbrt(beta))) - ((4.0 / (alpha * alpha)) - ((2.0 / alpha) + (8.0 / pow(alpha, 3.0))))) / 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 < 2.9257281358558516e23

    1. Initial program 0.8

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

    3. Applied add-sqr-sqrt_binary640.8

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

    4. Applied *-un-lft-identity_binary640.8

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

    5. Applied times-frac_binary640.8

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

    6. Simplified0.8

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

    7. Simplified0.8

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

    if 2.9257281358558516e23 < alpha

    1. Initial program 51.3

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

    3. Applied div-sub_binary6451.3

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

    4. Applied associate-+l-_binary6449.6

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

    5. Simplified49.6

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

    7. Applied add-cube-cbrt_binary6449.7

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

    8. Applied associate-/l*_binary6449.7

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

    9. Taylor expanded around inf 18.1

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    10. Simplified18.1

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9257281358558516 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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))