Average Error: 16.0 → 6.8
Time: 3.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 104449142829.8675:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot e^{\log \left(\sqrt[3]{\left(\alpha + \beta\right) + 2}\right)}}{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}} \cdot \frac{\sqrt[3]{\left(\alpha + \beta\right) + 2}}{\sqrt[3]{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 104449142829.8675:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot e^{\log \left(\sqrt[3]{\left(\alpha + \beta\right) + 2}\right)}}{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}} \cdot \frac{\sqrt[3]{\left(\alpha + \beta\right) + 2}}{\sqrt[3]{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\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 104449142829.8675)
   (/
    (-
     (/
      1.0
      (*
       (/
        (*
         (cbrt (+ (+ alpha beta) 2.0))
         (exp (log (cbrt (+ (+ alpha beta) 2.0)))))
        (* (cbrt beta) (cbrt beta)))
       (/ (cbrt (+ (+ alpha beta) 2.0)) (cbrt beta))))
     (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))
    2.0)
   (/
    (-
     (/ 1.0 (/ (+ (+ alpha beta) 2.0) 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 <= 104449142829.8675) {
		tmp = ((1.0 / (((cbrt((alpha + beta) + 2.0) * exp(log(cbrt((alpha + beta) + 2.0)))) / (cbrt(beta) * cbrt(beta))) * (cbrt((alpha + beta) + 2.0) / cbrt(beta)))) - ((alpha / ((alpha + beta) + 2.0)) - 1.0)) / 2.0;
	} else {
		tmp = ((1.0 / (((alpha + beta) + 2.0) / 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 < 104449142829.86749

    1. Initial program 0.2

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

    3. Applied div-sub_binary640.2

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

    4. Applied associate-+l-_binary640.2

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

    5. Simplified0.2

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

    7. Applied clear-num_binary640.2

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

    9. Applied add-cube-cbrt_binary640.5

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

    10. Applied add-cube-cbrt_binary640.2

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

    11. Applied times-frac_binary640.2

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

    13. Applied add-exp-log_binary641.3

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

    if 104449142829.86749 < alpha

    1. Initial program 49.5

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

    3. Applied div-sub_binary6449.5

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

    4. Applied associate-+l-_binary6448.0

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

    5. Simplified48.0

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

    7. Applied clear-num_binary6448.0

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

    8. Taylor expanded around inf 18.4

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

    9. Simplified18.4

      \[\leadsto \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 104449142829.8675:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot e^{\log \left(\sqrt[3]{\left(\alpha + \beta\right) + 2}\right)}}{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}} \cdot \frac{\sqrt[3]{\left(\alpha + \beta\right) + 2}}{\sqrt[3]{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))