Average Error: 0.2 → 0.0
Time: 15.7s
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 2018365 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 11.2s)Debug log

start103.0ms

Algorithm
intervals

setup385.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize41.0ms

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))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

27.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
10.0ms
(+ (* a a) (* b b))
5.0ms
(pow (+ (* a a) (* b b)) 2)

series121.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

55.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
32.0ms
(pow (+ (* a a) (* b b)) 2)
19.0ms
(* 4 (* b b))
14.0ms
(+ (* a a) (* b b))

simplify700.0ms

Counts
19 → 57
Calls

19 calls. Slowest were:

134.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
98.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
94.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune721.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize27.0ms

Local error

Found 4 expressions with local error:

2.1b
(* (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 b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

rewrite69.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

45.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
19.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
3.0ms
(* (pow a 2) (pow b 2))

series98.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

35.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
35.0ms
(+ (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))) (* 4 (* b b)))
17.0ms
(* (pow a 2) (pow b 2))
11.0ms
(* 4 (* b b))

simplify2.0s

Counts
31 → 71
Calls

31 calls. Slowest were:

317.0ms
(* (* (exp (pow b 4)) (* (exp (pow a 4)) (exp (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
197.0ms
(* (* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))) (exp (* 4 (* b b))))
189.0ms
(* (exp (pow b 4)) (exp (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))

prune1.2s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

16.3b
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
16.3b
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
16.3b
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
2.1b
(* (pow a 2) (pow b 2))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

7.0ms
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
5.0ms
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
5.0ms
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))

series392.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

135.0ms
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
119.0ms
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
110.0ms
(cbrt (+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2))))))
27.0ms
(* (pow a 2) (pow b 2))

simplify3.4s

Counts
41 → 76
Calls

41 calls. Slowest were:

773.0ms
(cbrt (- (* (pow b 4) (pow b 4)) (* (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))))
467.0ms
(cbrt (- (* (pow b 4) (pow b 4)) (* (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))))
436.0ms
(cbrt (- (* (pow b 4) (pow b 4)) (* (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))))

prune1.7s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes109.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms