Average Error: 52.3 → 37.2
Time: 5.0m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.162348338359012 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\frac{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}{\left(\beta + \alpha\right) - i \cdot 2}} \cdot \frac{i \cdot \frac{i \cdot \left(\beta + \alpha\right) + \left(i \cdot i + \alpha \cdot \beta\right)}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) + i \cdot \left(-4 \cdot i\right)}}{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{-1}{4}}{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}} \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\frac{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}{\left(\beta + \alpha\right) - i \cdot 2}}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.162348338359012e+154

    1. Initial program 50.3

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

    3. Applied times-frac35.0

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

    4. Applied associate-/l*35.0

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}}\]
    5. Using strategy rm

    6. Applied flip-+35.0

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

    7. Applied associate-/r/35.0

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

    8. Applied add-sqr-sqrt35.0

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

    9. Applied times-frac35.0

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

    10. Applied *-un-lft-identity35.0

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

    11. Applied times-frac35.0

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

    12. Applied times-frac35.0

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

    13. Simplified35.0

      \[\leadsto \color{blue}{\frac{i \cdot \frac{\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \alpha \cdot \beta\right)}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(i \cdot -4\right) \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}}{\left(\alpha + \beta\right) - 2 \cdot i}}\]

    if 1.162348338359012e+154 < beta

    1. Initial program 62.5

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

    3. Applied times-frac55.8

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

    4. Applied associate-/l*55.8

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}}\]
    5. Using strategy rm

    6. Applied flip-+55.8

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

    7. Applied associate-/r/55.8

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

    8. Applied add-sqr-sqrt55.8

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

    9. Applied times-frac55.8

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

    10. Applied *-un-lft-identity55.8

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

    11. Applied times-frac55.8

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

    12. Applied times-frac55.8

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

    13. Simplified55.8

      \[\leadsto \color{blue}{\frac{i \cdot \frac{\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \alpha \cdot \beta\right)}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(i \cdot -4\right) \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}}{\left(\alpha + \beta\right) - 2 \cdot i}}\]

    14. Taylor expanded around 0 48.3

      \[\leadsto \frac{i \cdot \color{blue}{\frac{-1}{4}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}}{\left(\alpha + \beta\right) - 2 \cdot i}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification37.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.162348338359012 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\frac{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}{\left(\beta + \alpha\right) - i \cdot 2}} \cdot \frac{i \cdot \frac{i \cdot \left(\beta + \alpha\right) + \left(i \cdot i + \alpha \cdot \beta\right)}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) + i \cdot \left(-4 \cdot i\right)}}{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{-1}{4}}{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}} \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\frac{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}{\left(\beta + \alpha\right) - i \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))

Details

Time bar (total: 5.0m)Debug log

sample80.0ms

Algorithm
intervals

simplify230.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

230.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 52.1b

localize71.0ms

Local error

Found 4 expressions with local error:

45.8b
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
34.8b
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
0.1b
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
0.0b
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

rewrite92.0ms

Algorithm
rewrite-expression-head
Counts
4 → 170
Calls

4 calls. Slowest were:

47.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
20.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
12.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

759.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
156.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
95.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
54.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify41.9s

Counts
191 → 182
Calls

191 calls. Slowest were:

838.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
696.0ms
(* (+ (pow (+ alpha beta) 3) (pow (* 2 i) 3)) (+ (pow (+ alpha beta) 3) (pow (* 2 i) 3)))
675.0ms
(/ (* (* (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))) (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))) (* (* (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))

prune2.6s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 34.7b

localize13.0ms

Local error

Found 4 expressions with local error:

34.2b
(/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))
33.7b
(/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))
28.6b
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i))))
4.9b
(/ (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i))) (/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))))

rewrite195.0ms

Algorithm
rewrite-expression-head
Counts
4 → 668
Calls

4 calls. Slowest were:

54.0ms
(/ (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i))) (/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))))
51.0ms
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i))))
39.0ms
(/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

731.0ms
(/ (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i))) (/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))))
518.0ms
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i))))
95.0ms
(/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))
57.0ms
(/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))

simplify3.3m

Counts
1143 → 680
Calls

1143 calls. Slowest were:

1.1s
(/ (sqrt (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
944.0ms
(/ (/ i (sqrt (+ (+ alpha beta) (* 2 i)))) (/ (sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (/ 1 (sqrt (+ (+ alpha beta) (* 2 i))))))
878.0ms
(/ (sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (/ (sqrt (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (cbrt (+ (+ alpha beta) (* 2 i))) (cbrt (+ (+ alpha beta) (* 2 i))))))

prune11.8s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 34.6b

localize84.0ms

Local error

Found 4 expressions with local error:

39.3b
(/ (+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha))) (+ (* beta (+ alpha i)) (* (+ alpha i) i)))
37.1b
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (- (+ alpha beta) (* 2 i)))
33.7b
(/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))
8.8b
(+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha)))

rewrite89.0ms

Algorithm
rewrite-expression-head
Counts
4 → 103
Calls

4 calls. Slowest were:

34.0ms
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (- (+ alpha beta) (* 2 i)))
28.0ms
(/ (+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha))) (+ (* beta (+ alpha i)) (* (+ alpha i) i)))
16.0ms
(/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))

series350.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

116.0ms
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (- (+ alpha beta) (* 2 i)))
92.0ms
(/ (+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha))) (+ (* beta (+ alpha i)) (* (+ alpha i) i)))
78.0ms
(/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))
63.0ms
(+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha)))

simplify20.0s

Counts
93 → 115
Calls

93 calls. Slowest were:

695.0ms
(+ (pow (* (* i i) -4) 3) (pow (* (+ beta alpha) (+ beta alpha)) 3))
662.0ms
(/ (* (cbrt (+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha)))) (cbrt (+ (* (* i i) -4) (* (+ beta alpha) (+ beta alpha))))) (* (cbrt (+ (* beta (+ alpha i)) (* (+ alpha i) i))) (cbrt (+ (* beta (+ alpha i)) (* (+ alpha i) i)))))
628.0ms
(/ (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))) (sqrt (- (+ alpha beta) (* 2 i))))

prune2.3s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 34.6b

localize76.0ms

Local error

Found 4 expressions with local error:

39.4b
(/ (+ (* (+ alpha beta) i) (+ (* i i) (* alpha beta))) (+ (* (+ alpha beta) (+ alpha beta)) (* (* i -4) i)))
37.1b
(sqrt (- (* (+ (+ alpha beta) (* i 2)) (+ (+ alpha beta) (* i 2))) 1.0))
37.1b
(sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
8.8b
(+ (* (+ alpha beta) (+ alpha beta)) (* (* i -4) i))

rewrite115.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

53.0ms
(/ (+ (* (+ alpha beta) i) (+ (* i i) (* alpha beta))) (+ (* (+ alpha beta) (+ alpha beta)) (* (* i -4) i)))
28.0ms
(sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
24.0ms
(sqrt (- (* (+ (+ alpha beta) (* i 2)) (+ (+ alpha beta) (* i 2))) 1.0))

series419.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
119.0ms
(sqrt (- (* (+ (+ alpha beta) (* i 2)) (+ (+ alpha beta) (* i 2))) 1.0))
104.0ms
(/ (+ (* (+ alpha beta) i) (+ (* i i) (* alpha beta))) (+ (* (+ alpha beta) (+ alpha beta)) (* (* i -4) i)))
64.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (* (* i -4) i))

simplify11.6s

Counts
60 → 88
Calls

60 calls. Slowest were:

561.0ms
(/ 1 (sqrt (+ (* (+ alpha beta) (+ alpha beta)) (* (* i -4) i))))
553.0ms
(sqrt (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))))
546.0ms
(sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))

prune1.9s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 34.6b

regimes206.0ms

Accuracy

51.5% (1.2b remaining)

Error of 37.2b against oracle of 36.0b and baseline of 38.4b

bsearch93.0ms

end0.0ms

sample2.5s

Algorithm
intervals