Average Error: 52.4 → 37.5
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.1297397571292906 \cdot 10^{+138}:\\ \;\;\;\;\frac{\sqrt{\left(\frac{i}{\alpha + (i \cdot 2 + \beta)_*} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + (i \cdot 2 + \beta)_*}\right) \cdot (\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left(\alpha + (i \cdot 2 + \beta)_*\right) \cdot \left(\alpha + (i \cdot 2 + \beta)_*\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\frac{i}{\alpha + (i \cdot 2 + \beta)_*} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + (i \cdot 2 + \beta)_*}\right) \cdot (\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left(\alpha + (i \cdot 2 + \beta)_*\right) \cdot \left(\alpha + (i \cdot 2 + \beta)_*\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.1297397571292906e+138

    1. Initial program 50.2

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

    3. Applied associate-/l*35.0

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

    5. Applied add-sqr-sqrt35.0

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

    6. Applied add-sqr-sqrt35.0

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

    7. Applied times-frac35.0

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

    8. Simplified35.0

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

    9. Simplified35.0

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

    if 5.1297397571292906e+138 < alpha

    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. Taylor expanded around -inf 48.6

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

  4. Final simplification37.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.1297397571292906 \cdot 10^{+138}:\\ \;\;\;\;\frac{\sqrt{\left(\frac{i}{\alpha + (i \cdot 2 + \beta)_*} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + (i \cdot 2 + \beta)_*}\right) \cdot (\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left(\alpha + (i \cdot 2 + \beta)_*\right) \cdot \left(\alpha + (i \cdot 2 + \beta)_*\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\frac{i}{\alpha + (i \cdot 2 + \beta)_*} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + (i \cdot 2 + \beta)_*}\right) \cdot (\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left(\alpha + (i \cdot 2 + \beta)_*\right) \cdot \left(\alpha + (i \cdot 2 + \beta)_*\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +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: 2.1m)Debug log

sample54.0ms

Algorithm
intervals
Results
256×(pre true 80)
256×(body real 80)

simplify221.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
221.0ms
(/ (/ (* (* 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: 51.1b

localize63.0ms

Local error

Found 4 expressions with local error:

45.3b
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
32.3b
(/ (/ (* (* 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)))

rewrite115.0ms

Algorithm
rewrite-expression-head
Rules
48×flip-+
48×flip3-+
44×times-frac
42×associate-/r/
38×add-exp-log
38×add-cbrt-cube
32×frac-times
28×associate-*r/
25×add-cube-cbrt
25×*-un-lft-identity
25×add-sqr-sqrt
16×associate-*l/
14×associate-/l*
13×div-exp
13×cbrt-undiv
11×pow1
10×cbrt-unprod
10×prod-exp
9×associate-/l/
6×div-inv
6×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×associate-*l*
4×associate-/r*
4×expm1-log1p-u
2×*-commutative
2×pow-prod-down
2×distribute-lft-in
2×frac-2neg
2×clear-num
2×distribute-rgt-in
1×flip--
1×pow-prod-up
1×flip3--
1×pow-plus
1×pow2
Counts
4 → 178
Calls
4 calls:
Slowest
45.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
44.0ms
(/ (/ (* (* 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))
12.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))
12.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
773.0ms
(/ (/ (* (* 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))
150.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
83.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
40.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify44.1s

Counts
191 → 190
Calls
191 calls:
Slowest
877.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
778.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (pow (+ alpha beta) 3) (pow (* 2 i) 3)) (+ (+ alpha beta) (* 2 i)))) (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))))
732.0ms
(* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i)))
697.0ms
(* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i)))
691.0ms
(/ (- (+ alpha beta) (* 2 i)) (sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))

prune3.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 32.2b

localize11.0ms

Local error

Found 4 expressions with local error:

36.7b
(/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
32.3b
(/ (/ (* 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))
4.7b
(/ (* 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)))

rewrite116.0ms

Algorithm
rewrite-expression-head
Rules
108×times-frac
39×add-cube-cbrt
39×*-un-lft-identity
39×add-sqr-sqrt
34×add-exp-log
34×add-cbrt-cube
21×associate-/r/
20×flip-+
20×flip3-+
16×associate-/l*
16×div-exp
16×cbrt-undiv
15×associate-/r*
13×div-inv
11×associate-/l/
9×pow1
8×frac-times
6×associate-*r/
4×add-log-exp
4×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×associate-*l/
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
3×frac-2neg
3×clear-num
1×flip--
1×pow-prod-up
1×*-commutative
1×pow-prod-down
1×flip3--
1×pow-plus
1×distribute-lft-in
1×distribute-rgt-in
1×pow2
Counts
4 → 187
Calls
4 calls:
Slowest
47.0ms
(/ (/ (* 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))
28.0ms
(/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
23.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))
13.0ms
(/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
712.0ms
(/ (/ (* 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))
160.0ms
(/ (* i (+ (+ alpha beta) i)) (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
133.0ms
(/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
40.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify43.0s

Counts
208 → 199
Calls
208 calls:
Slowest
917.0ms
(* (+ (+ alpha beta) (* 2 i)) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
890.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
861.0ms
(/ (/ i (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) 1)
786.0ms
(/ i (* (cbrt (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))) (cbrt (/ (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))))
772.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))

prune3.1s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 32.2b

localize30.0ms

Local error

Found 4 expressions with local error:

32.3b
(/ (* i (/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
31.3b
(/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha)))
0.0b
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))
0.0b
(* i (/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha))))

rewrite84.0ms

Algorithm
rewrite-expression-head
Rules
17×add-exp-log
17×add-cbrt-cube
12×add-cube-cbrt
12×*-un-lft-identity
12×add-sqr-sqrt
10×associate-*r*
9×times-frac
9×pow1
6×flip-+
6×flip3-+
6×associate-/r*
5×associate-*r/
5×div-exp
5×cbrt-undiv
4×add-log-exp
4×log1p-expm1-u
4×frac-times
4×expm1-log1p-u
3×div-inv
3×cbrt-unprod
3×prod-exp
3×associate-*l*
2×associate-/l/
2×associate-/l*
2×*-commutative
2×associate-/r/
2×associate-*l/
2×frac-2neg
2×clear-num
1×flip--
1×pow-prod-up
1×pow-prod-down
1×flip3--
1×pow-plus
1×distribute-lft-in
1×distribute-rgt-in
1×pow2
Counts
4 → 100
Calls
4 calls:
Slowest
40.0ms
(/ (* i (/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
22.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))
16.0ms
(/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha)))
3.0ms
(* i (/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha))))

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
762.0ms
(/ (* i (/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
178.0ms
(* i (/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha))))
127.0ms
(/ (* (fma i (+ i (+ beta alpha)) (* alpha beta)) (/ (+ i (+ beta alpha)) (fma 2 i (+ beta alpha)))) (fma 2 i (+ beta alpha)))
57.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify14.4s

Counts
76 → 112
Calls
76 calls:
Slowest
921.0ms
(/ i (* (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)) (cbrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))))
741.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))
608.0ms
(sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
588.0ms
(- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)
555.0ms
(/ i (sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))

prune1.8s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 32.2b

localize92.0ms

Local error

Found 4 expressions with local error:

35.2b
(sqrt (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0)))
35.2b
(sqrt (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0)))
26.5b
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha)))))
26.5b
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha)))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
8×sqrt-prod
8×associate-*r/
8×add-sqr-sqrt
6×add-cube-cbrt
6×sqrt-div
6×*-un-lft-identity
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×pow1/2
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
2×associate-*l/
2×rem-sqrt-square
2×frac-times
Counts
4 → 56
Calls
4 calls:
Slowest
4.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha)))))
4.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha)))))
1.0ms
(sqrt (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0)))
1.0ms
(sqrt (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0)))

series1.4s

Counts
4 → 12
Calls
4 calls:
Slowest
576.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha)))))
548.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 beta) alpha)) (/ i (+ (fma i 2 beta) alpha)))))
130.0ms
(sqrt (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0)))
103.0ms
(sqrt (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0)))

simplify4.5s

Counts
28 → 68
Calls
28 calls:
Slowest
656.0ms
(+ (* 2 (* (sqrt -1.0) (* i beta))) (+ (sqrt (- 1.0)) (* 2 (* (sqrt -1.0) (pow i 2)))))
639.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (+ (+ alpha i) beta) (/ i (+ (fma i 2 beta) alpha)))))
615.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (+ (+ alpha i) beta) (/ i (+ (fma i 2 beta) alpha)))))
513.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (+ (+ alpha i) beta) i)))
473.0ms
(sqrt (* (fma (+ (+ alpha i) beta) i (* beta alpha)) (* (+ (+ alpha i) beta) i)))

prune1.2s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 32.2b

regimes316.0ms

Accuracy

43.1% (1.6b remaining)

Error of 37.5b against oracle of 35.9b and baseline of 38.7b

bsearch108.0ms

end0.0ms

sample2.1s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)