Average Error: 3.6 → 0.2
Time: 5.1s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\frac{\left(\frac{\sqrt{\beta + 1}}{\left(\beta + \alpha\right) + 2} \cdot \sqrt{1 + \alpha}\right) \cdot \left(\frac{\sqrt{\beta + 1}}{\left(\beta + \alpha\right) + 2} \cdot \sqrt{1 + \alpha}\right)}{\alpha + \left(\beta + 3\right)}\]
\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}
↓
\frac{\left(\frac{\sqrt{\beta + 1}}{\left(\beta + \alpha\right) + 2} \cdot \sqrt{1 + \alpha}\right) \cdot \left(\frac{\sqrt{\beta + 1}}{\left(\beta + \alpha\right) + 2} \cdot \sqrt{1 + \alpha}\right)}{\alpha + \left(\beta + 3\right)}
(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
 (/
  (*
   (* (/ (sqrt (+ beta 1.0)) (+ (+ beta alpha) 2.0)) (sqrt (+ 1.0 alpha)))
   (* (/ (sqrt (+ beta 1.0)) (+ (+ beta alpha) 2.0)) (sqrt (+ 1.0 alpha))))
  (+ 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) {
	return (((sqrt(beta + 1.0) / ((beta + alpha) + 2.0)) * sqrt(1.0 + alpha)) * ((sqrt(beta + 1.0) / ((beta + alpha) + 2.0)) * sqrt(1.0 + alpha))) / (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 3.6

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

  2. Simplified2.2

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

  4. Applied add-sqr-sqrt_binary642.3

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

  5. Applied times-frac_binary640.2

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

  6. Applied add-sqr-sqrt_binary640.2

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

  7. Applied unswap-sqr_binary640.2

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

  8. Simplified0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020232 
(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)))