Average Error: 0.2 → 0.0
Time: 13.1s
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 2019022 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 12.6s)Debug log

sample94.0ms

Algorithm
intervals
Results
75.0ms661×body80valid

simplify341.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
341.0ms
(- (+ (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

localize18.0ms

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))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
8×add-log-exp
5×add-cube-cbrt
5×add-exp-log
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×add-cbrt-cube
3×unpow-prod-down
2×flip-+
2×flip3-+
2×sum-log
2×+-commutative
1×unpow2
1×*-commutative
1×pow-exp
1×associate-*r*
1×pow-pow
1×pow-to-exp
Counts
4 → 45
Calls
4 calls:
Slowest
12.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
9.0ms
(+ (* a a) (* b b))
2.0ms
(pow (+ (* a a) (* b b)) 2)
1.0ms
(* 4 (* b b))

series97.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
44.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
27.0ms
(pow (+ (* a a) (* b b)) 2)
15.0ms
(+ (* a a) (* b b))
11.0ms
(* 4 (* b b))

simplify721.0ms

Counts
19 → 57
Calls
19 calls:
Slowest
145.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
91.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
89.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
85.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
84.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune574.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize53.0ms

Local error

Found 4 expressions with local error:

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

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
8×add-exp-log
6×add-log-exp
6×add-cube-cbrt
6×associate-*r*
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
4×prod-exp
4×associate-*l*
4×pow1
4×pow-to-exp
3×*-commutative
2×unpow2
1×flip-+
1×cbrt-unprod
1×pow-prod-down
1×flip3-+
1×sum-log
1×+-commutative
Counts
4 → 51
Calls
4 calls:
Slowest
14.0ms
(* (pow a 2) (pow b 2))
11.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))
8.0ms
(* 2 (* (pow a 2) (pow b 2)))
2.0ms
(* 4 (* b b))

series111.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
44.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))
28.0ms
(* (pow a 2) (pow b 2))
21.0ms
(* 2 (* (pow a 2) (pow b 2)))
19.0ms
(* 4 (* b b))

simplify879.0ms

Counts
25 → 63
Calls
25 calls:
Slowest
182.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))
147.0ms
(* (exp (* 2 (* (pow a 2) (pow b 2)))) (exp (pow a 4)))
92.0ms
(+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))
92.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))
66.0ms
(* 2 (* (pow a 2) (pow b 2)))

prune889.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize32.0ms

Local error

Found 4 expressions with local error:

17.8b
(cbrt (pow (+ (* a a) (* b b)) 2))
17.8b
(cbrt (pow (+ (* a a) (* b b)) 2))
17.8b
(cbrt (pow (+ (* a a) (* b b)) 2))
0.4b
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
35×cbrt-prod
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
15×unpow-prod-down
10×associate-*l*
10×associate-*r*
9×pow1
7×pow1/3
6×add-exp-log
5×unpow2
4×add-log-exp
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 81
Calls
4 calls:
Slowest
14.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
4.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
4.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
4.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))

series447.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
120.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
118.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
118.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
91.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))

simplify3.2s

Counts
61 → 93
Calls
61 calls:
Slowest
646.0ms
(* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2))
555.0ms
(* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2))
100.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
93.0ms
(cbrt (* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2))))
93.0ms
(+ (log (cbrt (pow (+ (* a a) (* b b)) 2))) (log (cbrt (pow (+ (* a a) (* b b)) 2))))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes316.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample3.6s

Algorithm
intervals
Results
3.1s20530×body80valid