Average Error: 3.8 → 0.1
Time: 9.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[[alpha, beta] = \mathsf{sort}([alpha, beta]) \\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
↓
\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{t_0}\right)\right) \cdot \frac{1 + \beta}{t_0}}{\alpha + \left(\beta + 3\right)}\right)\right) \end{array} \]
\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}
t_0 := 2 + \left(\alpha + \beta\right)\\
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{t_0}\right)\right) \cdot \frac{1 + \beta}{t_0}}{\alpha + \left(\beta + 3\right)}\right)\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
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (expm1
    (log1p
     (/
      (* (expm1 (log1p (/ (+ 1.0 alpha) t_0))) (/ (+ 1.0 beta) t_0))
      (+ alpha (+ beta 3.0)))))))
double code(double alpha, double beta) {
	return (((((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);
}
↓
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	return expm1(log1p((expm1(log1p((1.0 + alpha) / t_0)) * ((1.0 + beta) / t_0)) / (alpha + (beta + 3.0))));
}

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. Initial program 3.8

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

  2. Simplified2.1

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

  3. Applied *-un-lft-identity_binary642.1

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

  4. Applied times-frac_binary640.1

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

  5. Applied associate-*r*_binary640.1

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

  6. Simplified0.1

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

  7. Applied expm1-log1p-u_binary640.1

    \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}\right)\right)} \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)} \]

  8. Applied expm1-log1p-u_binary640.1

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

  9. Final simplification0.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022067 
(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)))