Average Error: 3.3 → 0.1
Time: 29.4s
Precision: binary64
Cost: 1600
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\alpha + \left(\beta + 3\right)}\]
\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{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\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
 (/
  (*
   (/ (+ 1.0 beta) (+ (+ beta alpha) 2.0))
   (/ (+ 1.0 alpha) (+ (+ beta alpha) 2.0)))
  (+ 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) {
	return (((1.0 + beta) / ((beta + alpha) + 2.0)) * ((1.0 + alpha) / ((beta + alpha) + 2.0))) / (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

Alternatives

Alternative 1
Error0.1
Cost1600
\[\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2}}{\alpha + \left(\beta + 3\right)}\]
Alternative 2
Error2.2
Cost1921
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.2627346299296735 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \left(1 + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \frac{1 + \alpha}{\alpha + 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
Alternative 3
Error3.9
Cost1793
\[\begin{array}{l} \mathbf{if}\;\beta \leq 838592779798.6923:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \frac{1 + \alpha}{\alpha + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
Alternative 4
Error4.2
Cost1986
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.315621273650693 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \frac{1}{\beta + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{elif}\;\alpha \leq 9.692440884075845 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{1}{\alpha + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
Alternative 5
Error4.7
Cost1665
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 45816472080.235504:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2} \cdot \frac{1}{\beta + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
Alternative 6
Error31.7
Cost1025
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 176610707.98778427:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
Alternative 7
Error31.5
Cost1025
\[\begin{array}{l} \mathbf{if}\;\beta \leq 30828809.215161942:\\ \;\;\;\;\frac{1 + \beta}{\alpha \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
Alternative 8
Error32.2
Cost769
\[\begin{array}{l} \mathbf{if}\;\beta \leq 117316696.61084253:\\ \;\;\;\;\frac{1 + \beta}{\alpha \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array}\]
Alternative 9
Error38.0
Cost769
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.589968514197575 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 10
Error45.0
Cost64
\[0\]
Alternative 11
Error58.0
Cost64
\[1\]

Error

Derivation

  1. Initial program 3.3

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

  2. Simplified2.0

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

  4. Applied *-un-lft-identity_binary64_14422.0

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

  5. Applied times-frac_binary64_14480.2

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

  6. Applied associate-*r*_binary64_13820.1

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2021044 
(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)))