Average Error: 0.2 → 0.0
Time: 18.9s
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 2018365 +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: 14.0s)Debug log

start151.0ms

Algorithm
intervals

setup828.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize22.0ms

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))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

15.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
4.0ms
(+ (* a a) (* b b))
3.0ms
(pow (+ (* a a) (* b b)) 2)

series101.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

38.0ms
(pow (+ (* a a) (* b b)) 2)
33.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
15.0ms
(+ (* a a) (* b b))
15.0ms
(* 4 (* b b))

simplify1.2s

Counts
25 → 73
Calls

25 calls. Slowest were:

189.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
142.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
142.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune961.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* 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)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

7.0ms
(+ (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))

series124.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

58.0ms
(+ (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (* 4 (* b b)))
30.0ms
(fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))
19.0ms
(* 4 (* b b))
17.0ms
(* (* a b) (* a b))

simplify1.8s

Counts
22 → 68
Calls

22 calls. Slowest were:

736.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
150.0ms
(* (exp (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))) (exp (* 4 (* b b))))
149.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune990.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize35.0ms

Local error

Found 4 expressions with local error:

9.7b
(sqrt (+ (* a a) (* b b)))
0.3b
(* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2))
0.0b
(* 4 (* b b))
0.0b
(+ (* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2)) (* 4 (* b b)))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

16.0ms
(* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2))
15.0ms
(+ (* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2)) (* 4 (* b b)))
4.0ms
(sqrt (+ (* a a) (* b b)))

series175.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(+ (* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2)) (* 4 (* b b)))
52.0ms
(* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2))
31.0ms
(sqrt (+ (* a a) (* b b)))
12.0ms
(* 4 (* b b))

simplify2.2s

Counts
36 → 81
Calls

36 calls. Slowest were:

652.0ms
(* (* (* (fma b b (* a a)) (fma b b (* a a))) (fma b b (* a a))) (* (* (pow (sqrt (+ (* a a) (* b b))) 2) (pow (sqrt (+ (* a a) (* b b))) 2)) (pow (sqrt (+ (* a a) (* b b))) 2)))
387.0ms
(* (exp (* (fma b b (* a a)) (pow (sqrt (+ (* a a) (* b b))) 2))) (exp (* 4 (* b b))))
158.0ms
(sqrt (- (* (* a a) (* a a)) (* (* b b) (* b b))))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

16.3b
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
16.3b
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
16.3b
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
0.4b
(* (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))) (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))) (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))))
1.0ms
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
1.0ms
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))

series468.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
120.0ms
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
111.0ms
(cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
110.0ms
(* (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))) (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))))

simplify2.2s

Counts
41 → 81
Calls

41 calls. Slowest were:

154.0ms
(* (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))))
105.0ms
(cbrt (cbrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))))
98.0ms
(cbrt (sqrt (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))))

prune1.4s

Pruning

3 alts after pruning (0 fresh and 3 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