Average Error: 0.2 → 0.0
Time: 13.5s
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 2019016 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 13.1s)Debug log

sample148.0ms

Algorithm
intervals
Results
674×(pre true 80)
674×(body real 80)

simplify300.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
300.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize23.0ms

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))

rewrite38.0ms

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
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))
2.0ms
(* 4 (* b b))

series120.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
45.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
38.0ms
(pow (+ (* a a) (* b b)) 2)
25.0ms
(+ (* a a) (* b b))
12.0ms
(* 4 (* b b))

simplify698.0ms

Counts
19 → 57
Calls
19 calls:
Slowest
122.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
110.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
103.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
95.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
73.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune590.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize35.0ms

Local error

Found 4 expressions with local error:

2.9b
(* (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))))

rewrite54.0ms

Algorithm
rewrite-expression-head
Rules
15×add-log-exp
8×add-exp-log
7×sum-log
6×add-cube-cbrt
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
5×associate-*r*
4×prod-exp
4×associate-*l*
4×pow1
4×pow-to-exp
2×unpow2
2×flip-+
2×*-commutative
2×flip3-+
2×+-commutative
1×cbrt-unprod
1×pow-prod-down
1×associate-+l+
Counts
4 → 56
Calls
4 calls:
Slowest
41.0ms
(+ (+ (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))))
2.0ms
(* 4 (* b b))

series102.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
38.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
27.0ms
(+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))
25.0ms
(* (pow a 2) (pow b 2))
12.0ms
(* 4 (* b b))

simplify1.6s

Counts
28 → 68
Calls
28 calls:
Slowest
266.0ms
(* (* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
263.0ms
(* (* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
188.0ms
(* (exp (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
152.0ms
(* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))
130.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (* 4 (pow b 2))))

prune947.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize20.0ms

Local error

Found 4 expressions with local error:

15.0b
(log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))))
3.9b
(exp (log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))
0.3b
(pow (+ (* a a) (* b b)) 2)
0.0b
(* 4 (* b b))

rewrite43.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×pow1
6×add-exp-log
6×log-prod
4×add-log-exp
4×add-cbrt-cube
4×exp-prod
4×log-div
3×exp-sum
3×unpow-prod-down
2×flip-+
2×log-pow
2×exp-diff
2×flip3-+
1×rem-exp-log
1×unpow2
1×*-commutative
1×pow-exp
1×associate-*r*
1×rem-log-exp
1×pow-pow
1×pow-to-exp
Counts
4 → 54
Calls
4 calls:
Slowest
19.0ms
(exp (log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))
16.0ms
(log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))))
5.0ms
(pow (+ (* a a) (* b b)) 2)
2.0ms
(* 4 (* b b))

series161.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
90.0ms
(log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))))
38.0ms
(exp (log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))
20.0ms
(pow (+ (* a a) (* b b)) 2)
12.0ms
(* 4 (* b b))

simplify3.4s

Counts
33 → 66
Calls
33 calls:
Slowest
291.0ms
(exp (log (sqrt (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))))))
281.0ms
(log (* (cbrt (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))) (cbrt (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))))))
271.0ms
(exp (* (cbrt (log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))))) (cbrt (log (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))))
269.0ms
(exp (log (* (cbrt (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))) (cbrt (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))))))
261.0ms
(exp (log (+ (pow (pow (+ (* a a) (* b b)) 2) 3) (pow (* 4 (* b b)) 3))))

prune812.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes177.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample3.8s

Algorithm
intervals
Results
20583×(pre true 80)
20583×(body real 80)