Average Error: 0.2 → 0.0
Time: 10.9s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + (\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot 2 + \left({a}^{4} + {b}^{4}\right))_*\right) - 1\]

Error

Bits error versus a

Bits error versus b

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. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 10.6s)Debug log

sample156.0ms

Algorithm
intervals

simplify461.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

461.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize43.0ms

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 → 61
Calls

4 calls. Slowest were:

31.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
6.0ms
(pow (+ (* a a) (* b b)) 2)
4.0ms
(+ (* a a) (* b b))

series137.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
33.0ms
(pow (+ (* a a) (* b b)) 2)
25.0ms
(+ (* a a) (* b b))
19.0ms
(* 4 (* b b))

simplify1.6s

Counts
25 → 73
Calls

25 calls. Slowest were:

579.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
152.0ms
(sqrt (pow (+ (* a a) (* b b)) 2))
133.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune798.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* a b) (* a b))
0.0b
(* 4 (* b b))
0.0b
(+ (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (* 4 (* b b)))
0.0b
(fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

4.0ms
(+ (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (* 4 (* b b)))
3.0ms
(* (* a b) (* a b))
1.0ms
(* 4 (* b b))

series131.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(+ (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (* 4 (* b b)))
28.0ms
(* (* a b) (* a b))
27.0ms
(fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))
19.0ms
(* 4 (* b b))

simplify2.0s

Counts
22 → 68
Calls

22 calls. Slowest were:

769.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
180.0ms
(* (exp (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))) (exp (* 4 (* b b))))
156.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune923.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes255.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample4.0s

Algorithm
intervals