Average Error: 0.2 → 0.0
Time: 19.4s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + \left(\left({a}^{4} + 2 \cdot e^{\log \left({a}^{2} \cdot {b}^{2}\right)}\right) + {b}^{4}\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  3. Using strategy rm

  4. Applied add-exp-log0.0

    \[\leadsto \left(\left({b}^{4} + \left(2 \cdot \color{blue}{e^{\log \left({a}^{2} \cdot {b}^{2}\right)}} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  5. Final simplification0.0

    \[\leadsto \left(4 \cdot \left(b \cdot b\right) + \left(\left({a}^{4} + 2 \cdot e^{\log \left({a}^{2} \cdot {b}^{2}\right)}\right) + {b}^{4}\right)\right) - 1\]

Reproduce

herbie shell --seed 2019005 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 18.8s)Debug log

sample167.0ms

Algorithm
intervals

simplify378.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

378.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize20.0ms

Local error

Found 4 expressions with local error:

0.4b
(pow (+ (* a a) (* b b)) 2)
0.0b
(* 4 (* b b))
0.0b
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
0.0b
(+ (* a a) (* b b))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

26.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
4.0ms
(+ (* a a) (* b b))
3.0ms
(pow (+ (* a a) (* b b)) 2)

series115.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

45.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
32.0ms
(pow (+ (* a a) (* b b)) 2)
23.0ms
(* 4 (* b b))
15.0ms
(+ (* a a) (* b b))

simplify729.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

126.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
100.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
98.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))

prune611.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

2.7b
(* (pow a 2) (pow b 2))
0.0b
(* 4 (* b b))
0.0b
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
0.0b
(+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

46.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
7.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))
4.0ms
(* (pow a 2) (pow b 2))

series104.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
37.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))
17.0ms
(* (pow a 2) (pow b 2))
11.0ms
(* 4 (* b b))

simplify1.6s

Counts
28 → 68
Calls

28 calls. Slowest were:

258.0ms
(* (* (exp (pow b 4)) (* (exp (* 2 (* (pow a 2) (pow b 2)))) (exp (pow a 4)))) (exp (* 4 (* b b))))
223.0ms
(* (* (exp (pow b 4)) (exp (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))
176.0ms
(* (exp (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))

prune1.1s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize42.0ms

Local error

Found 4 expressions with local error:

38.3b
(log (* (pow a 2) (pow b 2)))
2.7b
(* (pow a 2) (pow b 2))
2.1b
(exp (log (* (pow a 2) (pow b 2))))
0.0b
(* 4 (* b b))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

10.0ms
(exp (log (* (pow a 2) (pow b 2))))
7.0ms
(* (pow a 2) (pow b 2))
6.0ms
(log (* (pow a 2) (pow b 2)))

series131.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(log (* (pow a 2) (pow b 2)))
21.0ms
(exp (log (* (pow a 2) (pow b 2))))
21.0ms
(* 4 (* b b))
17.0ms
(* (pow a 2) (pow b 2))

simplify708.0ms

Counts
36 → 72
Calls

36 calls. Slowest were:

157.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))
143.0ms
(- (+ (* 2 (log (/ -1 a))) (* 2 (log (/ -1 b)))))
88.0ms
(- (+ (* 2 (log (/ 1 b))) (* 2 (log (/ 1 a)))))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

31.0b
(cbrt (* (* (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))) (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))
0.4b
(pow (+ (* a a) (* b b)) 2)
0.4b
(pow (+ (* a a) (* b b)) 2)
0.4b
(pow (+ (* a a) (* b b)) 2)

rewrite79.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

69.0ms
(cbrt (* (* (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))) (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))
3.0ms
(pow (+ (* a a) (* b b)) 2)
3.0ms
(pow (+ (* a a) (* b b)) 2)

series122.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(cbrt (* (* (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))) (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))
32.0ms
(pow (+ (* a a) (* b b)) 2)
31.0ms
(pow (+ (* a a) (* b b)) 2)
20.0ms
(pow (+ (* a a) (* b b)) 2)

simplify6.7s

Counts
55 → 90
Calls

55 calls. Slowest were:

921.0ms
(cbrt (* (* (- (* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2)) (* (* 4 (* b b)) (* 4 (* b b)))) (- (* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2)) (* (* 4 (* b b)) (* 4 (* b b))))) (- (* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2)) (* (* 4 (* b b)) (* 4 (* b b))))))
443.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
235.0ms
(cbrt (* (* (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) (+ (pow (pow (+ (* a a) (* b b)) 2) 3) (pow (* 4 (* b b)) 3))) (+ (pow (pow (+ (* a a) (* b b)) 2) 3) (pow (* 4 (* b b)) 3))))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes158.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch5.0ms

end0.0ms

sample3.6s

Algorithm
intervals