Average Error: 0.2 → 0.0
Time: 10.7s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + (2 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right) - 1\]

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

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

Reproduce

herbie shell --seed 2019008 +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: 10.3s)Debug log

sample114.0ms

Algorithm
intervals

simplify485.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

485.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize39.0ms

Local error

Found 3 expressions with local error:

0.3b
(pow (+ (* a a) (* b b)) 2)
0.0b
(* 4 (* b b))
0.0b
(+ (* a a) (* b b))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 41
Calls

3 calls. Slowest were:

5.0ms
(pow (+ (* a a) (* b b)) 2)
4.0ms
(+ (* a a) (* b b))
2.0ms
(* 4 (* b b))

series77.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

39.0ms
(pow (+ (* a a) (* b b)) 2)
20.0ms
(* 4 (* b b))
18.0ms
(+ (* a a) (* b b))

simplify430.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

141.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
92.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
89.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune460.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize37.0ms

Local error

Found 4 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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

series80.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

28.0ms
(fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))
21.0ms
(+ (pow a 4) (pow b 4))
20.0ms
(* (* a b) (* a b))
11.0ms
(* 4 (* b b))

simplify666.0ms

Counts
22 → 68
Calls

22 calls. Slowest were:

406.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
70.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
64.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune762.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

13.1b
(sqrt (pow a 4))
13.1b
(sqrt (pow a 4))
0.2b
(fma (sqrt (pow a 4)) (sqrt (pow a 4)) (pow b 4))
0.1b
(* (* a b) (* a b))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

4.0ms
(* (* a b) (* a b))
1.0ms
(sqrt (pow a 4))
1.0ms
(sqrt (pow a 4))

series87.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

45.0ms
(fma (sqrt (pow a 4)) (sqrt (pow a 4)) (pow b 4))
24.0ms
(* (* a b) (* a b))
9.0ms
(sqrt (pow a 4))
9.0ms
(sqrt (pow a 4))

simplify611.0ms

Counts
26 → 68
Calls

26 calls. Slowest were:

520.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
20.0ms
(+ (log (* a b)) (log (* a b)))
14.0ms
(* (* a b) (* a b))

prune683.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

8.3b
(cbrt (fma b b (* a a)))
8.3b
(cbrt (+ (* a a) (* b b)))
0.5b
(pow (cbrt (+ (* a a) (* b b))) 2)
0.3b
(* (fma b b (* a a)) (cbrt (fma b b (* a a))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

9.0ms
(* (fma b b (* a a)) (cbrt (fma b b (* a a))))
3.0ms
(pow (cbrt (+ (* a a) (* b b))) 2)
1.0ms
(cbrt (+ (* a a) (* b b)))

series444.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

121.0ms
(* (fma b b (* a a)) (cbrt (fma b b (* a a))))
115.0ms
(cbrt (+ (* a a) (* b b)))
105.0ms
(pow (cbrt (+ (* a a) (* b b))) 2)
103.0ms
(cbrt (fma b b (* a a)))

simplify800.0ms

Counts
42 → 83
Calls

42 calls. Slowest were:

384.0ms
(cbrt (- (* (* a a) (* a a)) (* (* b b) (* b b))))
113.0ms
(* (* (* (fma b b (* a a)) (fma b b (* a a))) (fma b b (* a a))) (fma b b (* a a)))
36.0ms
(cbrt (+ (pow (* a a) 3) (pow (* b b) 3)))

prune1.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes151.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample3.2s

Algorithm
intervals