Average Error: 0.2 → 0.0
Time: 11.3s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + \left(\left({a}^{4} + \left({b}^{2} \cdot {a}^{2}\right) \cdot 2\right) + {b}^{4}\right)\right) - 1\]

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 10.9s)Debug log

sample154.0ms

Algorithm
intervals

simplify291.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

291.0ms
(- (+ (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

localize44.0ms

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))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

26.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
5.0ms
(pow (+ (* a a) (* b b)) 2)
4.0ms
(+ (* a a) (* b b))

series115.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

44.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
33.0ms
(pow (+ (* a a) (* b b)) 2)
21.0ms
(+ (* a a) (* b b))
17.0ms
(* 4 (* b b))

simplify716.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

120.0ms
(* (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)))))
90.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune533.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize55.0ms

Local error

Found 4 expressions with local error:

1.3b
(* (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))))) (* 4 (* b b)))
0.0b
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

43.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
8.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
6.0ms
(* (pow a 2) (pow b 2))

series136.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

59.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
44.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
17.0ms
(* (pow a 2) (pow b 2))
16.0ms
(* 4 (* b b))

simplify2.0s

Counts
31 → 71
Calls

31 calls. Slowest were:

318.0ms
(* (* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
233.0ms
(* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2))))))
181.0ms
(* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))

prune949.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

9.8b
(sqrt (+ (* a a) (* b b)))
0.4b
(* (+ (* a a) (* b b)) (pow (sqrt (+ (* a a) (* b b))) 2))
0.0b
(* 4 (* b b))
0.0b
(pow (sqrt (+ (* a a) (* b b))) 2)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

10.0ms
(* (+ (* a a) (* b b)) (pow (sqrt (+ (* a a) (* b b))) 2))
3.0ms
(pow (sqrt (+ (* a a) (* b b))) 2)
2.0ms
(sqrt (+ (* a a) (* b b)))

series95.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

31.0ms
(pow (sqrt (+ (* a a) (* b b))) 2)
29.0ms
(* (+ (* a a) (* b b)) (pow (sqrt (+ (* a a) (* b b))) 2))
19.0ms
(sqrt (+ (* a a) (* b b)))
15.0ms
(* 4 (* b b))

simplify1.2s

Counts
45 → 80
Calls

45 calls. Slowest were:

398.0ms
(* (+ (pow (* a a) 3) (pow (* b b) 3)) (pow (sqrt (+ (* a a) (* b b))) 2))
127.0ms
(sqrt (- (* (* a a) (* a a)) (* (* b b) (* b b))))
94.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune867.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes124.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample3.4s

Algorithm
intervals