Average Error: 52.3 → 29.2
Time: 3.0m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 6.487082572191457 \cdot 10^{+151}:\\ \;\;\;\;\left(\frac{\left(i + \beta\right) \cdot \frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \beta)_* + \alpha}}{\frac{(\left((2 \cdot i + \beta)_* + \alpha\right) \cdot \left((2 \cdot i + \beta)_* + \alpha\right) + \left(-1.0\right))_*}{i + \alpha}} \cdot \sqrt{\frac{i}{(2 \cdot i + \beta)_* + \alpha}}\right) \cdot \sqrt{\frac{i}{(2 \cdot i + \beta)_* + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{(2 \cdot i + \beta)_* + \alpha} \cdot \frac{\left(i + \beta\right) \cdot \frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \beta)_* + \alpha}}{(\beta \cdot 2 + \left((i \cdot 3 + \alpha)_*\right))_*}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 6.487082572191457e+151

    1. Initial program 42.2

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

    3. Applied associate-/l*14.8

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

    5. Applied *-un-lft-identity14.8

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

    6. Applied times-frac15.0

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

    7. Applied times-frac14.9

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

    8. Simplified14.9

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

    9. Simplified14.8

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

    11. Applied *-un-lft-identity14.8

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

    12. Applied times-frac14.8

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

    13. Simplified14.8

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

    14. Simplified9.8

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

    16. Applied add-sqr-sqrt10.3

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

    17. Applied associate-*l*10.3

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

    if 6.487082572191457e+151 < 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 associate-/l*61.6

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

    5. Applied *-un-lft-identity61.6

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

    6. Applied times-frac61.6

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

    7. Applied times-frac61.6

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

    8. Simplified61.6

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

    9. Simplified61.6

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

    11. Applied *-un-lft-identity61.6

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

    12. Applied times-frac61.6

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

    13. Simplified61.6

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

    14. Simplified61.4

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

    15. Taylor expanded around 0 47.3

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

    16. Simplified47.3

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

  4. Final simplification29.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.487082572191457 \cdot 10^{+151}:\\ \;\;\;\;\left(\frac{\left(i + \beta\right) \cdot \frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \beta)_* + \alpha}}{\frac{(\left((2 \cdot i + \beta)_* + \alpha\right) \cdot \left((2 \cdot i + \beta)_* + \alpha\right) + \left(-1.0\right))_*}{i + \alpha}} \cdot \sqrt{\frac{i}{(2 \cdot i + \beta)_* + \alpha}}\right) \cdot \sqrt{\frac{i}{(2 \cdot i + \beta)_* + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{(2 \cdot i + \beta)_* + \alpha} \cdot \frac{\left(i + \beta\right) \cdot \frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \beta)_* + \alpha}}{(\beta \cdot 2 + \left((i \cdot 3 + \alpha)_*\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +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: 3.0m)Debug log

start113.0ms

Algorithm
intervals

setup264.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 51.0b

localize50.0ms

Local error

Found 4 expressions with local error:

44.3b
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
38.0b
(/ (/ (* (* 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)))

rewrite133.0ms

Algorithm
rewrite-expression-head
Counts
4 → 178
Calls

4 calls. Slowest were:

57.0ms
(/ (/ (* (* 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))
25.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
23.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

764.0ms
(/ (/ (* (* 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))
154.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
88.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
39.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify49.1s

Counts
191 → 190
Calls

191 calls. Slowest were:

1.3s
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
922.0ms
(* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i)))
795.0ms
(* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i)))

prune3.7s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 37.9b

localize9.0ms

Local error

Found 4 expressions with local error:

41.1b
(/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
38.0b
(/ (/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
3.5b
(/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
0.0b
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

rewrite99.0ms

Algorithm
rewrite-expression-head
Counts
4 → 187
Calls

4 calls. Slowest were:

57.0ms
(/ (/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
14.0ms
(/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
13.0ms
(/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

740.0ms
(/ (/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
165.0ms
(/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
152.0ms
(/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
38.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify47.3s

Counts
208 → 199
Calls

208 calls. Slowest were:

1.1s
(/ (/ i (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) 1)
1.1s
(/ (/ i 1) (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))))
1.0s
(* (+ (+ alpha beta) (* 2 i)) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))

prune3.5s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 37.9b

localize51.0ms

Local error

Found 4 expressions with local error:

38.0b
(/ (* (/ i (+ (fma i 2 beta) alpha)) (* (+ (* alpha (+ beta i)) (* i (+ beta i))) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha))))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
3.7b
(* (/ i (+ (fma i 2 beta) alpha)) (* (+ (* alpha (+ beta i)) (* i (+ beta i))) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))))
0.0b
(/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))
0.0b
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

rewrite97.0ms

Algorithm
rewrite-expression-head
Counts
4 → 171
Calls

4 calls. Slowest were:

48.0ms
(/ (* (/ i (+ (fma i 2 beta) alpha)) (* (+ (* alpha (+ beta i)) (* i (+ beta i))) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha))))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
28.0ms
(* (/ i (+ (fma i 2 beta) alpha)) (* (+ (* alpha (+ beta i)) (* i (+ beta i))) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))))
12.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

762.0ms
(/ (* (/ i (+ (fma i 2 beta) alpha)) (* (+ (* alpha (+ beta i)) (* i (+ beta i))) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha))))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
175.0ms
(* (/ i (+ (fma i 2 beta) alpha)) (* (+ (* alpha (+ beta i)) (* i (+ beta i))) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))))
48.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))
44.0ms
(/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))

simplify33.0s

Counts
164 → 183
Calls

164 calls. Slowest were:

1.0s
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
1.0s
(sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
944.0ms
(* (+ (fma i 2 beta) alpha) (* (* (- beta i) (- beta i)) (fma 2 i (+ beta alpha))))

prune3.9s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 37.9b

localize31.0ms

Local error

Found 4 expressions with local error:

35.5b
(/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))
4.1b
(/ (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i)))
0.1b
(* (/ i (+ (fma 2 i beta) alpha)) (/ (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))))
0.0b
(/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 179
Calls

4 calls. Slowest were:

21.0ms
(* (/ i (+ (fma 2 i beta) alpha)) (/ (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))))
13.0ms
(/ (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i)))
12.0ms
(/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))

series1.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

880.0ms
(/ (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i)))
714.0ms
(* (/ i (+ (fma 2 i beta) alpha)) (/ (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))))
72.0ms
(/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))
46.0ms
(/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))

simplify27.6s

Counts
175 → 191
Calls

175 calls. Slowest were:

707.0ms
(/ i (- (* (fma 2 i beta) (fma 2 i beta)) (* alpha alpha)))
692.0ms
(/ (* (* (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha))) (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha)))) (* (+ beta i) (/ (+ (+ alpha i) beta) (+ (fma 2 i beta) alpha)))) (* (* (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i)) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))) (/ (fma (+ (fma 2 i beta) alpha) (+ (fma 2 i beta) alpha) (- 1.0)) (+ alpha i))))
684.0ms
(/ i (+ (pow (fma 2 i beta) 3) (pow alpha 3)))

prune3.3s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 26.0b

regimes251.0ms

Accuracy

60.8% (4.6b remaining)

Error of 29.2b against oracle of 24.6b and baseline of 36.4b

bsearch143.0ms