Average Error: 0.2 → 0.0
Time: 9.8s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + \left(\left({a}^{4} + \left({b}^{2} \cdot {a}^{2}\right) \cdot 2\right) + {b}^{4}\right)\right) - 1\]

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 2019007 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 9.3s)Debug log

sample126.0ms

Algorithm
intervals

simplify357.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

357.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize30.0ms

Local error

Found 4 expressions with local error:

0.3b
(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))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

22.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
4.0ms
(+ (* a a) (* b b))
3.0ms
(pow (+ (* a a) (* b b)) 2)

series110.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

41.0ms
(pow (+ (* a a) (* b b)) 2)
34.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
20.0ms
(* 4 (* b b))
15.0ms
(+ (* a a) (* b b))

simplify811.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

143.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
116.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
104.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune613.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize55.0ms

Local error

Found 4 expressions with local error:

2.2b
(* (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 a 4) (* 2 (* (pow a 2) (pow b 2))))

rewrite83.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

69.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
7.0ms
(* (pow a 2) (pow b 2))
4.0ms
(+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))

series120.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

44.0ms
(+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))
41.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
18.0ms
(* 4 (* b b))
16.0ms
(* (pow a 2) (pow b 2))

simplify1.6s

Counts
28 → 68
Calls

28 calls. Slowest were:

277.0ms
(* (* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
276.0ms
(* (* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
173.0ms
(* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))

prune1.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes237.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample4.0s

Algorithm
intervals