Average Error: 52.3 → 12.4
Time: 5.3m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 2.1349378046021964 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{(2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{i}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot \frac{\left(\alpha + \beta\right) + i}{(\left(\frac{\alpha}{i} \cdot \frac{\beta}{i}\right) \cdot -2 + 4)_*}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 2.1349378046021964e+100

    1. Initial program 31.9

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

    3. Applied times-frac13.0

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

    4. Applied associate-/l*13.0

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

    if 2.1349378046021964e+100 < i

    1. Initial program 62.1

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

    3. Applied times-frac50.6

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

    4. Applied associate-/l*50.6

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

    6. Applied add-sqr-sqrt50.7

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

    7. Applied *-un-lft-identity50.7

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

    8. Applied times-frac50.7

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

    9. Applied *-un-lft-identity50.7

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

    10. Applied times-frac50.7

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

    11. Applied add-sqr-sqrt50.8

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

    12. Applied times-frac50.8

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

    13. Applied times-frac50.8

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

    14. Simplified50.7

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

    15. Simplified50.6

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

    16. Taylor expanded around -inf 13.6

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

    17. Simplified12.6

      \[\leadsto \frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{i + \left(\beta + \alpha\right)}{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot \color{blue}{(\left(\frac{\beta}{i} \cdot \frac{\alpha}{i}\right) \cdot -2 + 4)_*}}\]
    18. Using strategy rm

    19. Applied *-un-lft-identity12.6

      \[\leadsto \frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\color{blue}{1 \cdot \left(i + \left(\beta + \alpha\right)\right)}}{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (\left(\frac{\beta}{i} \cdot \frac{\alpha}{i}\right) \cdot -2 + 4)_*}\]

    20. Applied times-frac12.7

      \[\leadsto \frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \color{blue}{\left(\frac{1}{(2 \cdot i + \left(\beta + \alpha\right))_*} \cdot \frac{i + \left(\beta + \alpha\right)}{(\left(\frac{\beta}{i} \cdot \frac{\alpha}{i}\right) \cdot -2 + 4)_*}\right)}\]

    21. Applied associate-*r*12.1

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.1349378046021964 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{(2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{i}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot \frac{\left(\alpha + \beta\right) + i}{(\left(\frac{\alpha}{i} \cdot \frac{\beta}{i}\right) \cdot -2 + 4)_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(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: 5.3m)Debug log

sample43.0ms

Algorithm
intervals

simplify174.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

174.0ms
(/ (/ (* (* 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

localize46.0ms

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)))

rewrite105.0ms

Algorithm
rewrite-expression-head
Counts
4 → 178
Calls

4 calls. Slowest were:

50.0ms
(/ (/ (* (* 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))
26.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
13.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series906.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

644.0ms
(/ (/ (* (* 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))
155.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
74.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
33.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify45.2s

Counts
191 → 190
Calls

191 calls. Slowest were:

996.0ms
(/ (- (+ alpha beta) (* 2 i)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))
945.0ms
(* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i)))
812.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))

prune3.0s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 34.7b

localize23.0ms

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)))))

rewrite129.0ms

Algorithm
rewrite-expression-head
Counts
4 → 676
Calls

4 calls. Slowest were:

45.0ms
(/ (/ (* 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)))))
31.0ms
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i))))
23.0ms
(/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

739.0ms
(/ (/ (* 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)))))
511.0ms
(/ (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0) (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i))))
88.0ms
(/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))
61.0ms
(/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i)))

simplify3.6m

Counts
1143 → 688
Calls

1143 calls. Slowest were:

1.0s
(/ 1 (/ (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))) (/ 1 1)))
979.0ms
(/ (/ i 1) (/ (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))) (/ 1 (sqrt (+ (+ alpha beta) (* 2 i))))))
931.0ms
(/ (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (/ (cbrt (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (sqrt (+ (+ alpha beta) (* 2 i)))))

prune16.6s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 34.6b

localize42.0ms

Local error

Found 4 expressions with local error:

38.6b
(/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))
8.8b
(/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))))
0.1b
(* (/ i (+ (fma i 2 beta) alpha)) (/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))))
0.0b
(* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

32.0ms
(* (/ i (+ (fma i 2 beta) alpha)) (/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))))
11.0ms
(* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))
11.0ms
(/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))))

series1.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

785.0ms
(* (/ i (+ (fma i 2 beta) alpha)) (/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))))
588.0ms
(/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))))
197.0ms
(* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))
169.0ms
(/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))

simplify18.5s

Counts
101 → 137
Calls

101 calls. Slowest were:

693.0ms
(* (* (* (/ i (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha))) (/ i (+ (fma i 2 beta) alpha))) (/ (* (* (+ i (+ beta alpha)) (+ i (+ beta alpha))) (+ i (+ beta alpha))) (* (* (* (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha))) (fma 2 i (+ beta alpha))) (/ (* (* (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0))) (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0))) (* (* (fma (+ i (+ beta alpha)) i (* alpha beta)) (fma (+ i (+ beta alpha)) i (* alpha beta))) (fma (+ i (+ beta alpha)) i (* alpha beta)))))))
656.0ms
(/ i (- (* (fma i 2 beta) (fma i 2 beta)) (* alpha alpha)))
625.0ms
(* (* (* (/ i (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha))) (/ i (+ (fma i 2 beta) alpha))) (/ (* (* (+ i (+ beta alpha)) (+ i (+ beta alpha))) (+ i (+ beta alpha))) (* (* (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta))))) (* (fma 2 i (+ beta alpha)) (/ (fma (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha)) (- 1.0)) (fma (+ i (+ beta alpha)) i (* alpha beta)))))))

prune2.5s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 8.3b

localize21.0ms

Local error

Found 4 expressions with local error:

2.0b
(* (/ beta i) (/ alpha i))
1.9b
(/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (fma (* (/ beta i) (/ alpha i)) -2 4)))
0.0b
(* (/ i (+ (fma i 2 beta) alpha)) (/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (fma (* (/ beta i) (/ alpha i)) -2 4))))
0.0b
(/ i (+ (fma i 2 beta) alpha))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

10.0ms
(* (/ i (+ (fma i 2 beta) alpha)) (/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (fma (* (/ beta i) (/ alpha i)) -2 4))))
8.0ms
(/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (fma (* (/ beta i) (/ alpha i)) -2 4)))
4.0ms
(* (/ beta i) (/ alpha i))

series304.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(* (/ i (+ (fma i 2 beta) alpha)) (/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (fma (* (/ beta i) (/ alpha i)) -2 4))))
94.0ms
(/ (+ i (+ beta alpha)) (* (fma 2 i (+ beta alpha)) (fma (* (/ beta i) (/ alpha i)) -2 4)))
59.0ms
(/ i (+ (fma i 2 beta) alpha))
24.0ms
(* (/ beta i) (/ alpha i))

simplify8.1s

Counts
66 → 111
Calls

66 calls. Slowest were:

731.0ms
(/ i (- (* (fma i 2 beta) (fma i 2 beta)) (* alpha alpha)))
578.0ms
(* (* (* (/ i (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha))) (/ i (+ (fma i 2 beta) alpha))) (/ (* (* (+ i (+ beta alpha)) (+ i (+ beta alpha))) (+ i (+ beta alpha))) (* (* (* (fma 2 i (+ beta alpha)) (fma 2 i (+ beta alpha))) (fma 2 i (+ beta alpha))) (* (* (fma (* (/ beta i) (/ alpha i)) -2 4) (fma (* (/ beta i) (/ alpha i)) -2 4)) (fma (* (/ beta i) (/ alpha i)) -2 4)))))
574.0ms
(/ i (+ (pow (fma i 2 beta) 3) (pow alpha 3)))

prune1.7s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 8.3b

regimes372.0ms

Accuracy

42.3% (2.5b remaining)

Error of 12.4b against oracle of 9.9b and baseline of 14.2b

bsearch81.0ms

end0.0ms

sample2.4s

Algorithm
intervals