Average Error: 0.2 → 0.0
Time: 24.9s
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 2018362 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 20.0s)Debug log

start182.0ms

Algorithm
intervals

setup374.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize39.0ms

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))

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

26.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
12.0ms
(pow (+ (* a a) (* b b)) 2)
4.0ms
(+ (* a a) (* b b))

series94.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

45.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
23.0ms
(pow (+ (* a a) (* b b)) 2)
15.0ms
(+ (* a a) (* b b))
11.0ms
(* 4 (* b b))

simplify868.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

162.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
156.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
122.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune952.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize32.0ms

Local error

Found 4 expressions with local error:

1.8b
(* (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
(* 2 (* (pow a 2) (pow b 2)))

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

38.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
4.0ms
(* 2 (* (pow a 2) (pow b 2)))
3.0ms
(* (pow a 2) (pow b 2))

series101.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

40.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
22.0ms
(* 2 (* (pow a 2) (pow b 2)))
19.0ms
(* 4 (* b b))
19.0ms
(* (pow a 2) (pow b 2))

simplify1.4s

Counts
28 → 66
Calls

28 calls. Slowest were:

276.0ms
(* (* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
272.0ms
(* (* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
166.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))

prune1.2s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

32.0b
(cbrt (* (* (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))))
1.8b
(* (pow a 2) (pow b 2))
1.8b
(* (pow a 2) (pow b 2))
1.8b
(* (pow a 2) (pow b 2))

rewrite170.0ms

Algorithm
rewrite-expression-head
Counts
4 → 102
Calls

4 calls. Slowest were:

159.0ms
(cbrt (* (* (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))))
3.0ms
(* (pow a 2) (pow b 2))
3.0ms
(* (pow a 2) (pow b 2))

series120.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(cbrt (* (* (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))))
29.0ms
(* (pow a 2) (pow b 2))
19.0ms
(* (pow a 2) (pow b 2))
17.0ms
(* (pow a 2) (pow b 2))

simplify11.8s

Counts
73 → 114
Calls

73 calls. Slowest were:

998.0ms
(cbrt (* (* (- (* (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (* (* 4 (* b b)) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))) (+ (pow (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) 3) (pow (* 4 (* b b)) 3))))
583.0ms
(cbrt (* (* (- (* (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (* (* 4 (* b b)) (* 4 (* b b)))) (+ (pow (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) 3) (pow (* 4 (* b b)) 3))) (+ (pow (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) 3) (pow (* 4 (* b b)) 3))))
566.0ms
(cbrt (* (* (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b))) (+ (pow (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) 3) (pow (* 4 (* b b)) 3))) (+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))))

prune2.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes102.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms