Average Error: 0.2 → 0.0
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right)\right) - 1\]

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 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. Final simplification0.0

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

Reproduce

herbie shell --seed 2018360 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 14.1s)Debug log

start150.0ms

Algorithm
intervals

setup277.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize20.0ms

Local error

Found 4 expressions with local error:

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

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

13.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
4.0ms
(+ (* a a) (* b b))
3.0ms
(pow (+ (* a a) (* b b)) 2)

series107.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

40.0ms
(pow (+ (* a a) (* b b)) 2)
40.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
17.0ms
(+ (* a a) (* b b))
11.0ms
(* 4 (* b b))

simplify935.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

316.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
110.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
105.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune723.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

2.8b
(* (pow a 2) (pow b 2))
0.0b
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
0.0b
(- (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))) 1)
0.0b
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))

rewrite306.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

246.0ms
(- (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))) 1)
49.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
8.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))

series152.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
44.0ms
(- (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))) 1)
28.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
27.0ms
(* (pow a 2) (pow b 2))

simplify2.7s

Counts
33 → 75
Calls

33 calls. Slowest were:

343.0ms
(* (exp (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))
324.0ms
(* (* (exp (pow b 4)) (* (exp (* 2 (* (pow a 2) (pow b 2)))) (exp (pow a 4)))) (exp (* 4 (* b b))))
293.0ms
(* (* (exp (pow b 4)) (exp (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))

prune1.3s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

12.0b
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
12.0b
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
2.8b
(* (pow a 2) (pow b 2))
2.8b
(* (pow a 2) (pow b 2))

rewrite132.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

65.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
59.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
3.0ms
(* (pow a 2) (pow b 2))

series195.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

86.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
75.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
17.0ms
(* (pow a 2) (pow b 2))
17.0ms
(* (pow a 2) (pow b 2))

simplify5.0s

Counts
46 → 84
Calls

46 calls. Slowest were:

1.3s
(sqrt (+ (pow (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) 3) (pow (* 4 (* b b)) 3)))
858.0ms
(sqrt (+ (pow (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) 3) (pow (* 4 (* b b)) 3)))
373.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))

prune1.9s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes111.0ms

Accuracy

0% (0.0b remaining)

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

bsearch5.0ms