Average Error: 3.7 → 0.1
Time: 39.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[[alpha, beta]=\mathsf{sort}([alpha, beta])\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \beta}}\]
\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}
↓
\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \beta}}
(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
 (/
  (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
  (* (+ alpha (+ 2.0 beta)) (/ (+ alpha (+ beta 3.0)) (+ 1.0 beta)))))
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 ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((alpha + (2.0 + beta)) * ((alpha + (beta + 3.0)) / (1.0 + beta)));
}

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 3.7

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

  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 *-un-lft-identity_binary64_7602.2

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

  5. Applied times-frac_binary64_7660.1

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

  6. Applied associate-*r*_binary64_7000.1

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

  7. Simplified0.1

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

  9. Applied associate-/l*_binary64_7050.1

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

  10. Simplified0.1

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

  12. Applied associate-/r/_binary64_7060.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2021075 
(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)))