Average Error: 4.0 → 3.7
Time: 3.5m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.0313287992006405 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{\frac{1}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \sqrt{\left(1.0 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}\right) \cdot \frac{\sqrt{\frac{\left(\beta + \left(1.0 + \alpha\right)\right) + \beta \cdot \alpha}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{\frac{2.0}{\alpha}}{\alpha}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

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 < 5.0313287992006405e+31

    1. Initial program 0.2

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.8

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    4. Applied add-sqr-sqrt0.3

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    5. Applied times-frac0.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Simplified0.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Simplified0.3

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \color{blue}{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity0.3

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    10. Applied sqrt-prod0.3

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\color{blue}{\sqrt{1} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    11. Applied div-inv0.3

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\color{blue}{\left(\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}}}{\sqrt{1} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    12. Applied sqrt-prod0.3

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\color{blue}{\sqrt{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)} \cdot \sqrt{\frac{1}{2 + \left(\beta + \alpha\right)}}}}{\sqrt{1} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    13. Applied times-frac0.2

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \color{blue}{\left(\frac{\sqrt{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}}{\sqrt{1}} \cdot \frac{\sqrt{\frac{1}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    14. Simplified0.2

      \[\leadsto \frac{\frac{\sqrt{\frac{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \left(\color{blue}{\sqrt{\left(1.0 + \alpha\right) + \left(\beta + \alpha \cdot \beta\right)}} \cdot \frac{\sqrt{\frac{1}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)}}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    if 5.0313287992006405e+31 < alpha

    1. Initial program 12.6

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Taylor expanded around inf 11.4

      \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    3. Simplified11.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2.0}{\alpha}}{\alpha} + \left(1 - \frac{1.0}{\alpha}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.0313287992006405 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{\frac{1}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \sqrt{\left(1.0 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}\right) \cdot \frac{\sqrt{\frac{\left(\beta + \left(1.0 + \alpha\right)\right) + \beta \cdot \alpha}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{\frac{2.0}{\alpha}}{\alpha}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))

Details

Time bar (total: 3.5m)Debug log

sample51.0ms

Algorithm
intervals
Results
32.0ms256×body80valid
6.0ms256×pre80true

simplify576.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
576.0ms
(/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

prune21.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 3.8b

localize52.0ms

Local error

Found 4 expressions with local error:

3.6b
(/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1)))
0.1b
(/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))
0.0b
(/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1)))
0.0b
(+ (+ alpha beta) (* beta alpha))

rewrite199.0ms

Algorithm
rewrite-expression-head
Rules
531×times-frac
256×add-cube-cbrt
256×*-un-lft-identity
256×add-sqr-sqrt
69×associate-/l*
46×associate-/r/
25×flip-+
25×flip3-+
23×div-inv
20×add-exp-log
20×add-cbrt-cube
10×div-exp
10×cbrt-undiv
9×associate-/r*
6×add-log-exp
4×associate-/l/
4×pow1
3×frac-2neg
3×clear-num
1×sum-log
1×associate-+l+
1×+-commutative
Counts
4 → 349
Calls
4 calls:
Slowest
80.0ms
(/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))
68.0ms
(/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1)))
36.0ms
(/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1)))
3.0ms
(+ (+ alpha beta) (* beta alpha))

series238.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
101.0ms
(/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))
78.0ms
(/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1)))
45.0ms
(/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1)))
14.0ms
(+ (+ alpha beta) (* beta alpha))

simplify1.4m

Counts
524 → 361
Calls
524 calls:
Slowest
922.0ms
(/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 1) (* 2 1)))) (sqrt (+ (+ alpha beta) (* 2 1)))) (* (cbrt (+ (+ (+ alpha beta) (* 2 1)) 1.0)) (cbrt (+ (+ (+ alpha beta) (* 2 1)) 1.0))))
724.0ms
(/ (/ (/ (cbrt (+ (+ (+ alpha beta) (* beta alpha)) 1.0)) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))
699.0ms
(/ (/ (/ (sqrt (+ (+ (+ alpha beta) (* beta alpha)) 1.0)) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))
683.0ms
(/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))
681.0ms
(/ (/ (/ (sqrt (+ (+ (+ alpha beta) (* beta alpha)) 1.0)) (+ (+ alpha beta) (* 2 1))) (sqrt (+ (+ alpha beta) (* 2 1)))) (cbrt (+ (+ (+ alpha beta) (* 2 1)) 1.0)))

prune5.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 1.4b

localize47.0ms

Local error

Found 4 expressions with local error:

3.6b
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
3.6b
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
0.3b
(* (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))) (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))
0.2b
(/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))

rewrite97.0ms

Algorithm
rewrite-expression-head
Rules
513×times-frac
450×sqrt-prod
284×add-cube-cbrt
284×*-un-lft-identity
284×add-sqr-sqrt
114×associate-*l*
114×associate-*r*
48×associate-/r/
26×flip-+
26×flip3-+
24×associate-/l*
24×div-inv
22×add-exp-log
22×add-cbrt-cube
12×associate-/r*
9×pow1
7×div-exp
7×sqrt-div
7×cbrt-undiv
5×associate-/l/
5×sqrt-undiv
4×add-log-exp
4×cbrt-unprod
4×prod-exp
3×frac-2neg
3×clear-num
1×pow-prod-up
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×frac-times
1×pow2
Counts
4 → 463
Calls
4 calls:
Slowest
25.0ms
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
25.0ms
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
24.0ms
(* (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))) (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))
12.0ms
(/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))

series336.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
124.0ms
(/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))
88.0ms
(* (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))) (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))
66.0ms
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
57.0ms
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))

simplify52.8s

Counts
561 → 475
Calls
561 calls:
Slowest
591.0ms
(+ (pow (* alpha beta) 3) (pow (+ beta (+ 1.0 alpha)) 3))
580.0ms
(+ (pow (* alpha beta) 3) (pow (+ beta (+ 1.0 alpha)) 3))
539.0ms
(/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (- (* 2 2) (* (+ beta alpha) (+ beta alpha))))) (sqrt (sqrt (+ 2 (+ beta alpha)))))
518.0ms
(/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (- (* 2 2) (* (+ beta alpha) (+ beta alpha))))) (sqrt (sqrt (+ 2 (+ beta alpha)))))
513.0ms
(sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (- (* 2 2) (* (+ beta alpha) (+ beta alpha)))))

prune9.2s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 1.4b

localize23.0ms

Local error

Found 4 expressions with local error:

3.6b
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
3.6b
(sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))
0.3b
(* (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))) (* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (/ (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))))
0.2b
(* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (/ (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))

rewrite63.0ms

Algorithm
rewrite-expression-head
Rules
225×times-frac
215×sqrt-prod
135×*-un-lft-identity
132×add-sqr-sqrt
131×add-cube-cbrt
121×associate-*l*
61×associate-*r*
51×flip-+
51×flip3-+
46×sqrt-div
32×add-exp-log
32×add-cbrt-cube
31×frac-times
30×associate-/r/
28×frac-add
19×associate-*l/
16×associate-*r/
15×div-inv
12×cbrt-unprod
12×prod-exp
11×pow1
7×div-exp
7×cbrt-undiv
4×add-log-exp
4×pow-prod-down
3×associate-/l*
3×sqrt-unprod
3×associate-/r*
3×sqrt-undiv
2×associate-/l/
2×distribute-lft-out
2×*-commutative
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
Counts
4 → 309
Calls
4 calls:
Slowest
31.0ms
(* (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))) (* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (/ (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))))
12.0ms
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
11.0ms
(* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (/ (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))
4.0ms
(sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))

series461.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
255.0ms
(* (/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))) (* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (/ (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))))
106.0ms
(* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (/ (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))
51.0ms
(/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))
49.0ms
(sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))

simplify37.8s

Counts
312 → 321
Calls
312 calls:
Slowest
924.0ms
(/ (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (- (* 2 2) (* (+ beta alpha) (+ beta alpha))))) (sqrt (sqrt (+ 2 (+ beta alpha)))))
686.0ms
(+ (pow (* alpha beta) 3) (pow (+ beta (+ 1.0 alpha)) 3))
606.0ms
(* (sqrt (- (+ 1.0 alpha) (+ beta (* alpha beta)))) (sqrt (+ 2 (+ beta alpha))))
575.0ms
(* (/ (* (* (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha)))) (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha))))) (sqrt (/ (+ (* alpha beta) (+ beta (+ 1.0 alpha))) (+ 2 (+ beta alpha))))) (* (* (sqrt (+ 2 (+ beta alpha))) (sqrt (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha))))) (* (* (* (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta)))) (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))) (sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))) (/ (* (* (sqrt (/ 1 (+ 2 (+ beta alpha)))) (sqrt (/ 1 (+ 2 (+ beta alpha))))) (sqrt (/ 1 (+ 2 (+ beta alpha))))) (* (* (sqrt (+ 2 (+ beta alpha))) (sqrt (+ 2 (+ beta alpha)))) (sqrt (+ 2 (+ beta alpha)))))))
543.0ms
(sqrt (* (+ (* 1.0 1.0) (- (* alpha alpha) (* 1.0 alpha))) (+ (* beta beta) (- (* (* alpha beta) (* alpha beta)) (* beta (* alpha beta))))))

prune6.9s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 1.4b

localize33.0ms

Local error

Found 4 expressions with local error:

3.9b
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
3.9b
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
3.9b
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
3.6b
(sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
10×*-un-lft-identity
9×add-sqr-sqrt
9×cbrt-prod
8×flip-+
8×add-cube-cbrt
8×flip3-+
6×sqrt-div
6×cbrt-div
4×add-log-exp
4×sqrt-prod
4×add-exp-log
4×frac-add
4×add-cbrt-cube
4×pow1
3×pow1/3
1×distribute-lft-out
1×pow1/2
1×rem-sqrt-square
Counts
4 → 58
Calls
4 calls:
Slowest
7.0ms
(sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))
4.0ms
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
4.0ms
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
4.0ms
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))

series196.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
58.0ms
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
47.0ms
(cbrt (+ (* alpha beta) (+ beta (+ 1.0 alpha))))
29.0ms
(sqrt (+ (+ 1.0 alpha) (+ beta (* alpha beta))))

simplify6.5s

Counts
38 → 70
Calls
38 calls:
Slowest
608.0ms
(cbrt (+ (pow (* alpha beta) 3) (pow (+ beta (+ 1.0 alpha)) 3)))
592.0ms
(sqrt (+ (* (+ (pow 1.0 3) (pow alpha 3)) (+ (* beta beta) (- (* (* alpha beta) (* alpha beta)) (* beta (* alpha beta))))) (* (+ (* 1.0 1.0) (- (* alpha alpha) (* 1.0 alpha))) (+ (pow beta 3) (pow (* alpha beta) 3)))))
517.0ms
(cbrt (+ (pow (* alpha beta) 3) (pow (+ beta (+ 1.0 alpha)) 3)))
498.0ms
(cbrt (+ (pow (* alpha beta) 3) (pow (+ beta (+ 1.0 alpha)) 3)))
445.0ms
(cbrt (- (* (* alpha beta) (* alpha beta)) (* (+ beta (+ 1.0 alpha)) (+ beta (+ 1.0 alpha)))))

prune1.5s

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 1.4b

regimes149.0ms

Accuracy

15.4% (2.3b remaining)

Error of 3.7b against oracle of 1.4b and baseline of 4.1b

bsearch113.0ms

end0.0ms

sample1.6s

Algorithm
intervals
Results
1.2s8000×body80valid
266.0ms8000×pre80true