Average Error: 0.2 → 0.0
Time: 25.2s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right) + \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) - 1\]

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. Using strategy rm

  4. Applied add-cube-cbrt0.0

    \[\leadsto \left(\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right) + \color{blue}{\left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}}\right) - 1\]

  5. Final simplification0.0

    \[\leadsto \left(\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right) + \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) - 1\]

Reproduce

herbie shell --seed 2019002 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 24.7s)Debug log

sample203.0ms

Algorithm
intervals

simplify235.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

235.0ms
(- (+ (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
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
0.0b
(* 4 (* b b))
0.0b
(+ (* a a) (* b b))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

13.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
2.0ms
(pow (+ (* a a) (* b b)) 2)
2.0ms
(+ (* a a) (* b b))

series82.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

34.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
20.0ms
(pow (+ (* a a) (* b b)) 2)
16.0ms
(+ (* a a) (* b b))
13.0ms
(* 4 (* b b))

simplify768.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

126.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
114.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
113.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune482.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 expressions with local error:

2.9b
(* (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
(* 4 (* b b))
0.0b
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))

rewrite52.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

40.0ms
(+ (+ (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)))
3.0ms
(* (pow a 2) (pow b 2))

series127.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

46.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
41.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
23.0ms
(* (pow a 2) (pow b 2))
17.0ms
(* 4 (* b b))

simplify2.1s

Counts
31 → 71
Calls

31 calls. Slowest were:

247.0ms
(* (* (exp (pow b 4)) (exp (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))
246.0ms
(* (* (exp (pow b 4)) (* (exp (* 2 (* (pow a 2) (pow b 2)))) (exp (pow a 4)))) (exp (* 4 (* b b))))
234.0ms
(* (exp (pow b 4)) (* (exp (* 2 (* (pow a 2) (pow b 2)))) (exp (pow a 4))))

prune1.0s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

25.9b
(cbrt (* 4 (* b b)))
25.9b
(cbrt (* 4 (* b b)))
25.9b
(cbrt (* 4 (* b b)))
2.9b
(* (pow a 2) (pow b 2))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

3.0ms
(* (pow a 2) (pow b 2))
1.0ms
(cbrt (* 4 (* b b)))
1.0ms
(cbrt (* 4 (* b b)))

series946.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

340.0ms
(cbrt (* 4 (* b b)))
290.0ms
(cbrt (* 4 (* b b)))
286.0ms
(cbrt (* 4 (* b b)))
30.0ms
(* (pow a 2) (pow b 2))

simplify302.0ms

Counts
26 → 61
Calls

26 calls. Slowest were:

134.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))
25.0ms
(+ (log (pow a 2)) (* (log b) 2))
25.0ms
(+ (* (log a) 2) (log (pow b 2)))

prune902.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

29.6b
(cbrt (* (* (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))))
2.9b
(* (pow a 2) (pow b 2))
2.9b
(* (pow a 2) (pow b 2))
2.9b
(* (pow a 2) (pow b 2))

rewrite263.0ms

Algorithm
rewrite-expression-head
Counts
4 → 102
Calls

4 calls. Slowest were:

252.0ms
(cbrt (* (* (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))))
3.0ms
(* (pow a 2) (pow b 2))
3.0ms
(* (pow a 2) (pow b 2))

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(cbrt (* (* (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))))
29.0ms
(* (pow a 2) (pow b 2))
28.0ms
(* (pow a 2) (pow b 2))
22.0ms
(* (pow a 2) (pow b 2))

simplify11.3s

Counts
73 → 114
Calls

73 calls. Slowest were:

658.0ms
(cbrt (* (* (- (* (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (* (* 4 (* b b)) (* 4 (* b b)))) (+ (pow (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) 3) (pow (* 4 (* b b)) 3))) (- (* (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (* (* 4 (* b b)) (* 4 (* b b))))))
613.0ms
(cbrt (* (* (- (* (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (* (* 4 (* b b)) (* 4 (* b b)))) (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))) (+ (pow (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) 3) (pow (* 4 (* b b)) 3))))
608.0ms
(cbrt (* (* (- (* (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (* (* 4 (* b b)) (* 4 (* b b)))) (+ (pow (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) 3) (pow (* 4 (* b b)) 3))) (+ (pow (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) 3) (pow (* 4 (* b b)) 3))))

prune2.0s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes61.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample3.6s

Algorithm
intervals