Average Error: 0.2 → 0.0
Time: 12.8s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(b \cdot b\right) \cdot 4 + \left(\left({\left(a \cdot b\right)}^{2} \cdot 2 + {a}^{4}\right) + {b}^{4}\right)\right) - 1\]

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 inf 0.0

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

  4. Applied pow-prod-down0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019008 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 12.4s)Debug log

sample113.0ms

Algorithm
intervals

simplify257.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

257.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 3 expressions with local error:

0.3b
(pow (+ (* a a) (* b b)) 2)
0.0b
(* 4 (* b b))
0.0b
(+ (* a a) (* b b))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 34
Calls

3 calls. Slowest were:

3.0ms
(pow (+ (* a a) (* b b)) 2)
2.0ms
(+ (* a a) (* b b))
1.0ms
(* 4 (* b b))

series47.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

19.0ms
(pow (+ (* a a) (* b b)) 2)
16.0ms
(+ (* a a) (* b b))
11.0ms
(* 4 (* b b))

simplify376.0ms

Counts
15 → 43
Calls

15 calls. Slowest were:

119.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
90.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
81.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))

prune421.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize50.0ms

Local error

Found 4 expressions with local error:

2.4b
(* (pow a 2) (pow b 2))
0.0b
(* 2 (* (pow a 2) (pow b 2)))
0.0b
(* 4 (* b b))
0.0b
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

10.0ms
(* (pow a 2) (pow b 2))
8.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
4.0ms
(* 2 (* (pow a 2) (pow b 2)))

series93.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

32.0ms
(* 2 (* (pow a 2) (pow b 2)))
26.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
19.0ms
(* (pow a 2) (pow b 2))
16.0ms
(* 4 (* b b))

simplify1.2s

Counts
28 → 66
Calls

28 calls. Slowest were:

221.0ms
(* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
203.0ms
(* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2))))))
139.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))

prune888.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.1b
(pow (* a b) 2)
0.0b
(* 2 (pow (* a b) 2))
0.0b
(* 4 (* b b))
0.0b
(+ (pow b 4) (+ (pow a 4) (* 2 (pow (* a b) 2))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

8.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (pow (* a b) 2))))
2.0ms
(* 2 (pow (* a b) 2))
1.0ms
(pow (* a b) 2)

series89.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

36.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (pow (* a b) 2))))
23.0ms
(* 2 (pow (* a b) 2))
19.0ms
(pow (* a b) 2)
11.0ms
(* 4 (* b b))

simplify1.5s

Counts
24 → 60
Calls

24 calls. Slowest were:

448.0ms
(* 2 (* (pow a 2) (pow b 2)))
258.0ms
(* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (pow (* a b) 2)))))
178.0ms
(* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (pow (* a b) 2)))))

prune855.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

7.6b
(sqrt (+ (* a a) (* b b)))
0.3b
(* (+ (* a a) (* b b)) (pow (sqrt (+ (* a a) (* b b))) 2))
0.0b
(pow (sqrt (+ (* a a) (* b b))) 2)
0.0b
(* 4 (* b b))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

24.0ms
(* (+ (* a a) (* b b)) (pow (sqrt (+ (* a a) (* b b))) 2))
5.0ms
(pow (sqrt (+ (* a a) (* b b))) 2)
4.0ms
(sqrt (+ (* a a) (* b b)))

series111.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(pow (sqrt (+ (* a a) (* b b))) 2)
32.0ms
(* (+ (* a a) (* b b)) (pow (sqrt (+ (* a a) (* b b))) 2))
30.0ms
(sqrt (+ (* a a) (* b b)))
11.0ms
(* 4 (* b b))

simplify1.4s

Counts
45 → 80
Calls

45 calls. Slowest were:

467.0ms
(* (+ (pow (* a a) 3) (pow (* b b) 3)) (pow (sqrt (+ (* a a) (* b b))) 2))
140.0ms
(* (- (* (* a a) (* a a)) (* (* b b) (* b b))) (pow (sqrt (+ (* a a) (* b b))) 2))
120.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune935.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes101.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample3.8s

Algorithm
intervals