Average Error: 23.2 → 7.2
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999944522127678:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}}\right) + 1.0}{2.0}\\ \end{array}\]

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 (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) < -0.9999944522127678

    1. Initial program 61.8

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around -inf 32.1

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified32.1

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}}{2.0}\]

    if -0.9999944522127678 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

    1. Initial program 12.1

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity12.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity12.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac0.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac0.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Simplified0.0

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    10. Applied *-un-lft-identity0.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    11. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    12. Applied times-frac0.1

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    13. Applied times-frac0.1

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]

    14. Applied associate-*r*0.1

      \[\leadsto \frac{\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999944522127678:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}}\right) + 1.0}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))

Details

Time bar (total: 1.3m)Debug log

sample66.0ms

Algorithm
intervals

simplify561.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
561.0ms
(/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0)

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 23.6b

localize26.0ms

Local error

Found 3 expressions with local error:

20.2b
(/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i)))
1.6b
(+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0)
0.1b
(/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

rewrite149.0ms

Algorithm
rewrite-expression-head
Rules
42×times-frac
23×add-cube-cbrt
23×*-un-lft-identity
23×add-sqr-sqrt
17×add-exp-log
17×add-cbrt-cube
12×associate-/r/
10×associate-/l*
10×flip-+
10×flip3-+
9×associate-/l/
7×div-exp
7×cbrt-undiv
6×div-inv
6×associate-/r*
4×frac-times
3×add-log-exp
3×flip--
3×flip3--
3×pow1
2×cbrt-unprod
2×associate-*r/
2×prod-exp
2×associate-*l/
2×frac-2neg
2×clear-num
1×+-commutative
Counts
3 → 99
Calls
3 calls:
Slowest
85.0ms
(+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0)
46.0ms
(/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
14.0ms
(/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i)))

series538.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
284.0ms
(+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0)
188.0ms
(/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
65.0ms
(/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i)))

simplify14.4s

Counts
107 → 108
Calls
107 calls:
Slowest
595.0ms
(/ (/ (* (* (* (+ alpha beta) (- beta alpha)) (* (+ alpha beta) (- beta alpha))) (* (+ alpha beta) (- beta alpha))) (* (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (+ alpha beta) (* 2 i)))) (* (* (+ (+ (+ alpha beta) (* 2 i)) 2.0) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
526.0ms
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
491.0ms
(/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
440.0ms
(/ (/ (* (+ alpha beta) (- beta alpha)) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i)))) 1)
439.0ms
(/ (/ 1 (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

prune1.4s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 5.9b

localize16.0ms

Local error

Found 4 expressions with local error:

3.6b
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
2.6b
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
0.0b
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
0.0b
(/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))

rewrite64.0ms

Algorithm
rewrite-expression-head
Rules
162×times-frac
94×add-cube-cbrt
94×*-un-lft-identity
94×add-sqr-sqrt
51×associate-*r*
20×associate-/r/
20×add-exp-log
20×add-cbrt-cube
18×associate-/l*
13×flip-+
13×flip3-+
10×div-inv
7×div-exp
7×cbrt-undiv
6×associate-/r*
6×pow1
4×add-log-exp
3×associate-/l/
3×cbrt-unprod
3×prod-exp
3×associate-*l*
3×div-sub
2×associate-*l/
2×frac-2neg
2×clear-num
2×frac-times
1×flip--
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×flip3--
1×+-commutative
Counts
4 → 197
Calls
4 calls:
Slowest
24.0ms
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
20.0ms
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
12.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
5.0ms
(/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))

series584.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
192.0ms
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
173.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
152.0ms
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
68.0ms
(/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))

simplify27.1s

Counts
224 → 209
Calls
224 calls:
Slowest
868.0ms
(+ (log (+ beta alpha)) (log (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))))
602.0ms
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
582.0ms
(* (- (* beta beta) (* alpha alpha)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
507.0ms
(* (+ (pow beta 3) (pow alpha 3)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
495.0ms
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

prune2.7s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 5.9b

localize27.0ms

Local error

Found 4 expressions with local error:

1.6b
(+ (/ (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0)
0.1b
(* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))
0.1b
(/ (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
0.0b
(/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))

rewrite99.0ms

Algorithm
rewrite-expression-head
Rules
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
21×times-frac
20×add-exp-log
20×add-cbrt-cube
15×associate-*r*
8×flip-+
8×flip3-+
7×associate-/l/
6×associate-/r/
6×div-exp
6×associate-/r*
6×cbrt-undiv
6×pow1
4×add-log-exp
4×associate-/l*
4×cbrt-unprod
4×prod-exp
4×associate-*l/
4×frac-times
3×div-inv
3×associate-*l*
2×associate-*r/
2×frac-2neg
2×clear-num
1×flip--
1×*-commutative
1×pow-prod-down
1×div-sub
1×flip3--
1×+-commutative
Counts
4 → 108
Calls
4 calls:
Slowest
57.0ms
(+ (/ (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0)
27.0ms
(/ (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
8.0ms
(* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))
5.0ms
(/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))

series436.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
174.0ms
(/ (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
158.0ms
(+ (/ (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0)
58.0ms
(* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))
47.0ms
(/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))

simplify6.7s

Counts
94 → 120
Calls
94 calls:
Slowest
427.0ms
(/ (* (* (* (+ beta alpha) (+ beta alpha)) (+ beta alpha)) (* (* (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))) (* (* (+ (+ (+ alpha beta) (* 2 i)) 2.0) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
394.0ms
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
374.0ms
(- (+ (log (+ beta alpha)) (log (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))) (log (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
347.0ms
(/ (* (* (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))) (* (+ beta alpha) (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))) (* (* (+ (+ (+ alpha beta) (* 2 i)) 2.0) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
326.0ms
(* (* (* (+ beta alpha) (+ beta alpha)) (+ beta alpha)) (/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) (+ (+ alpha beta) (* 2 i)))))

prune1.5s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 5.9b

localize32.0ms

Local error

Found 4 expressions with local error:

3.6b
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
3.6b
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
3.6b
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
2.6b
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))

rewrite74.0ms

Algorithm
rewrite-expression-head
Rules
72×times-frac
51×associate-*r*
42×add-cube-cbrt
42×*-un-lft-identity
42×add-sqr-sqrt
13×add-exp-log
13×add-cbrt-cube
9×flip-+
9×flip3-+
8×associate-/r/
6×pow1
4×add-log-exp
4×div-inv
3×cbrt-unprod
3×prod-exp
3×associate-*l*
3×div-exp
3×+-commutative
3×cbrt-undiv
2×associate-*l/
2×frac-times
1×*-commutative
1×associate-*r/
1×pow-prod-down
Counts
4 → 104
Calls
4 calls:
Slowest
23.0ms
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
21.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
15.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
13.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)

series623.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
171.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
170.0ms
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
142.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)
140.0ms
(+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)

simplify13.4s

Counts
79 → 116
Calls
79 calls:
Slowest
852.0ms
(* (- (* beta beta) (* alpha alpha)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))
515.0ms
(+ (log (+ beta alpha)) (log (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))))
494.0ms
(/ (/ (sqrt (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
474.0ms
(/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))
465.0ms
(* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))

prune2.1s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 5.9b

regimes518.0ms

Accuracy

75.6% (1.6b remaining)

Error of 7.2b against oracle of 5.6b and baseline of 12.1b

bsearch7.0ms

end0.0ms

sample3.0s

Algorithm
intervals