Average Error: 0.2 → 0.0
Time: 15.4s
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 2019022 +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: 15.1s)Debug log

sample89.0ms

Algorithm
intervals
Results
74.0ms661×body80valid

simplify478.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
478.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize39.0ms

Local error

Found 4 expressions with local error:

0.4b
(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))

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
8×add-log-exp
8×fma-def
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×unpow-prod-down
5×add-exp-log
5×pow1
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×unpow2
2×flip-+
2×flip3-+
2×sum-log
2×+-commutative
1×*-commutative
1×pow-exp
1×associate-*r*
1×pow-pow
1×pow-to-exp
Counts
4 → 61
Calls
4 calls:
Slowest
30.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
5.0ms
(pow (+ (* a a) (* b b)) 2)
4.0ms
(+ (* a a) (* b b))
2.0ms
(* 4 (* b b))

series94.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
36.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
25.0ms
(pow (+ (* a a) (* b b)) 2)
22.0ms
(+ (* a a) (* b b))
11.0ms
(* 4 (* b b))

simplify1.1s

Counts
25 → 73
Calls
25 calls:
Slowest
183.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
119.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
118.0ms
(sqrt (pow (+ (* a a) (* b b)) 2))
98.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
98.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune676.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize22.0ms

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×log1p-expm1-u
4×expm1-log1p-u
3×fma-def
2×fma-udef
2×*-commutative
2×associate-*r*
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×pow-plus
1×flip3-+
1×sum-log
1×associate-+l+
1×+-commutative
1×pow2
Counts
4 → 56
Calls
4 calls:
Slowest
4.0ms
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (* b b)))
3.0ms
(* (* a b) (* a b))
1.0ms
(* 4 (* b b))
0.0ms
(fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))

series96.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
37.0ms
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (* b b)))
30.0ms
(fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))
18.0ms
(* (* a b) (* a b))
11.0ms
(* 4 (* b b))

simplify2.0s

Counts
22 → 68
Calls
22 calls:
Slowest
901.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
166.0ms
(* (exp (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (exp (* 4 (* b b))))
141.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
99.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
98.0ms
(* 2 (* (* a b) (* a b)))

prune871.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

17.8b
(cbrt (pow (+ (* a a) (* b b)) 2))
17.8b
(cbrt (pow (+ (* a a) (* b b)) 2))
17.8b
(cbrt (pow (+ (* a a) (* b b)) 2))
0.4b
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))

rewrite37.0ms

Algorithm
rewrite-expression-head
Rules
35×cbrt-prod
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
15×unpow-prod-down
10×associate-*l*
10×associate-*r*
9×pow1
7×pow1/3
6×add-exp-log
5×unpow2
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 89
Calls
4 calls:
Slowest
24.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
4.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
4.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
4.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))

series418.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
114.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
110.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
107.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))
87.0ms
(cbrt (pow (+ (* a a) (* b b)) 2))

simplify4.5s

Counts
61 → 101
Calls
61 calls:
Slowest
765.0ms
(* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2))
740.0ms
(* (pow (+ (* a a) (* b b)) 2) (pow (+ (* a a) (* b b)) 2))
172.0ms
(cbrt (cbrt (pow (+ (* a a) (* b b)) 2)))
170.0ms
(sqrt (cbrt (pow (+ (* a a) (* b b)) 2)))
155.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes331.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample3.1s

Algorithm
intervals
Results
2.7s20530×body80valid