Average Error: 0.2 → 0.0
Time: 14.6s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + (2 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right) - 1\]

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 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. Simplified0.0

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

  4. Final simplification0.0

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

Runtime

Time bar (total: 14.6s)Debug logProfile

start250.0ms

Algorithm
intervals

setup578.0ms

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

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

16.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
6.0ms
(pow (+ (* a a) (* b b)) 2)
2.0ms
(+ (* a a) (* b b))

series107.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

45.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
33.0ms
(pow (+ (* a a) (* b b)) 2)
18.0ms
(+ (* a a) (* b b))

simplify1.4s

Counts
25 → 73
Calls

25 calls. Slowest were:

230.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
165.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
141.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune1.0s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize38.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* a b) (* a b))
0.0b
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (* b b)))
0.0b
(* 4 (* b b))
0.0b
(fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

59.0ms
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (* b b)))
31.0ms
(fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))
20.0ms
(* (* a b) (* a b))

simplify2.1s

Counts
22 → 68
Calls

22 calls. Slowest were:

730.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
248.0ms
(* (exp (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (exp (* 4 (* b b))))
178.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune1.1s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))