Average Error: 52.0 → 37.3
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.6738311742745447 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{(\left(\alpha + \beta\right) \cdot i + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\alpha + \beta\right))_*\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\alpha + i\right) \cdot \left(i + \beta\right)} \cdot \frac{\sqrt{\left(\alpha + i\right) \cdot \left(i + \beta\right)}}{(2 \cdot i + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) \cdot \left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.6738311742745447e+146

    1. Initial program 49.7

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

    3. Applied add-sqr-sqrt49.7

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

    4. Applied times-frac34.8

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

    5. Applied times-frac34.8

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

    6. Simplified34.8

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

    7. Simplified34.8

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

    9. Applied *-un-lft-identity34.8

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

    10. Applied add-sqr-sqrt34.8

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

    11. Applied times-frac34.8

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

    12. Simplified34.8

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

    13. Simplified34.8

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

    if 1.6738311742745447e+146 < 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 *-un-lft-identity62.5

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

    4. Applied times-frac54.6

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

    5. Applied times-frac54.6

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

    6. Simplified54.6

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

    7. Simplified54.6

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

    8. Taylor expanded around -inf 48.8

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification37.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.6738311742745447 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{(\left(\alpha + \beta\right) \cdot i + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\alpha + \beta\right))_*\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\alpha + i\right) \cdot \left(i + \beta\right)} \cdot \frac{\sqrt{\left(\alpha + i\right) \cdot \left(i + \beta\right)}}{(2 \cdot i + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) \cdot \left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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: 1.9m)Debug log

sample94.0ms

Algorithm
intervals

simplify265.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

265.0ms
(/ (/ (* (* 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))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 52.3b

localize36.0ms

Local error

Found 4 expressions with local error:

45.9b
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
36.9b
(/ (/ (* (* 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)))

rewrite163.0ms

Algorithm
rewrite-expression-head
Counts
4 → 178
Calls

4 calls. Slowest were:

73.0ms
(/ (/ (* (* 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))
38.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
23.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

729.0ms
(/ (/ (* (* 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))
192.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
108.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
44.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify45.3s

Counts
191 → 190
Calls

191 calls. Slowest were:

1.0s
(* (* (* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i))) (* i (+ (+ alpha beta) i))) (* (* (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
902.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
890.0ms
(* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i)))

prune3.0s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 36.8b

localize51.0ms

Local error

Found 4 expressions with local error:

38.4b
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
38.4b
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
35.8b
(/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma i 2 (+ alpha beta)))
35.5b
(/ (fma (+ alpha beta) i (* i i)) (fma 2 i (+ alpha beta)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

3.0ms
(/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma i 2 (+ alpha beta)))
3.0ms
(/ (fma (+ alpha beta) i (* i i)) (fma 2 i (+ alpha beta)))
1.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))

series368.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
98.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
73.0ms
(/ (fma (+ alpha beta) i (* i i)) (fma 2 i (+ alpha beta)))
70.0ms
(/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma i 2 (+ alpha beta)))

simplify8.9s

Counts
70 → 98
Calls

70 calls. Slowest were:

625.0ms
(sqrt (sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))))
623.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
613.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))

prune2.0s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 36.8b

localize46.0ms

Local error

Found 4 expressions with local error:

38.4b
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
38.4b
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
35.6b
(sqrt (* (+ alpha i) (+ i beta)))
35.6b
(sqrt (* (+ alpha i) (+ i beta)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

4.0ms
(sqrt (* (+ alpha i) (+ i beta)))
3.0ms
(sqrt (* (+ alpha i) (+ i beta)))
2.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))

series332.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

138.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
89.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
59.0ms
(sqrt (* (+ alpha i) (+ i beta)))
46.0ms
(sqrt (* (+ alpha i) (+ i beta)))

simplify7.9s

Counts
38 → 78
Calls

38 calls. Slowest were:

625.0ms
(sqrt (* (cbrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))) (cbrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))))
609.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
607.0ms
(sqrt (* (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))) (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))))

prune1.7s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 36.8b

localize37.0ms

Local error

Found 4 expressions with local error:

35.8b
(/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta)))
35.5b
(/ (fma i (+ alpha i) (* i beta)) (+ (fma i 2 beta) alpha))
30.6b
(/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
4.8b
(* (/ (fma i (+ alpha i) (* i beta)) (+ (fma i 2 beta) alpha)) (/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 216
Calls

4 calls. Slowest were:

29.0ms
(* (/ (fma i (+ alpha i) (* i beta)) (+ (fma i 2 beta) alpha)) (/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))
12.0ms
(/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
8.0ms
(/ (fma i (+ alpha i) (* i beta)) (+ (fma i 2 beta) alpha))

series1.5s

Counts
4 → 12
Calls

4 calls. Slowest were:

854.0ms
(* (/ (fma i (+ alpha i) (* i beta)) (+ (fma i 2 beta) alpha)) (/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))
500.0ms
(/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
82.0ms
(/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta)))
61.0ms
(/ (fma i (+ alpha i) (* i beta)) (+ (fma i 2 beta) alpha))

simplify32.5s

Counts
236 → 228
Calls

236 calls. Slowest were:

882.0ms
(/ (/ 1 (sqrt (fma i 2 (+ alpha beta)))) (sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))
679.0ms
(* (fma i (+ alpha i) (* i beta)) (/ (/ (fma (+ (+ alpha i) beta) i (* beta alpha)) (fma i 2 (+ alpha beta))) (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))
643.0ms
(/ (/ 1 1) (* (cbrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))) (cbrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))))

prune4.0s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 36.8b

regimes252.0ms

Accuracy

40.1% (1.6b remaining)

Error of 37.3b against oracle of 35.8b and baseline of 38.4b

bsearch146.0ms

end0.0ms

sample2.1s

Algorithm
intervals