Average Error: 0.2 → 0.0
Time: 12.9s
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 2019016 +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: 12.5s)Debug log

sample192.0ms

Algorithm
intervals
Results
674×(pre true 80)
674×(body real 80)

simplify519.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
519.0ms
(- (+ (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

localize19.0ms

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))

rewrite26.0ms

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
20.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
3.0ms
(pow (+ (* a a) (* b b)) 2)
2.0ms
(+ (* a a) (* b b))
1.0ms
(* 4 (* b b))

series97.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
42.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
20.0ms
(pow (+ (* a a) (* b b)) 2)
18.0ms
(+ (* a a) (* b b))
16.0ms
(* 4 (* b b))

simplify1.3s

Counts
25 → 73
Calls
25 calls:
Slowest
198.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
172.0ms
(* (cbrt (pow (+ (* a a) (* b b)) 2)) (cbrt (pow (+ (* a a) (* b b)) 2)))
140.0ms
(sqrt (pow (+ (* a a) (* b b)) 2))
127.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
118.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))

prune736.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize23.0ms

Local error

Found 3 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)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
8×pow1
5×add-log-exp
5×add-exp-log
5×add-cbrt-cube
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×log1p-expm1-u
3×fma-def
3×expm1-log1p-u
2×*-commutative
2×associate-*r*
1×fma-udef
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
3 → 46
Calls
3 calls:
Slowest
4.0ms
(* (* a b) (* a b))
4.0ms
(+ (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (* 4 (* b b)))
1.0ms
(* 4 (* b b))

series70.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
42.0ms
(+ (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4))) (* 4 (* b b)))
17.0ms
(* (* a b) (* a b))
11.0ms
(* 4 (* b b))

simplify1.5s

Counts
19 → 55
Calls
19 calls:
Slowest
657.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
151.0ms
(* (exp (fma (* (* a b) (* a b)) 2 (+ (pow b 4) (pow a 4)))) (exp (* 4 (* b b))))
142.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
133.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
108.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune675.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

2.1b
(expm1 (log1p (pow (+ (* a a) (* b b)) 2)))
0.3b
(pow (+ (* a a) (* b b)) 2)
0.0b
(* 4 (* b b))
0.0b
(+ (expm1 (log1p (pow (+ (* a a) (* b b)) 2))) (* 4 (* b b)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
6×add-log-exp
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×add-exp-log
5×pow1
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
3×fma-def
3×unpow-prod-down
2×expm1-udef
1×unpow2
1×flip-+
1×associate-+l-
1×*-commutative
1×pow-exp
1×associate-*r*
1×flip3-+
1×pow-pow
1×sum-log
1×expm1-log1p
1×+-commutative
1×pow-to-exp
Counts
4 → 55
Calls
4 calls:
Slowest
6.0ms
(+ (expm1 (log1p (pow (+ (* a a) (* b b)) 2))) (* 4 (* b b)))
5.0ms
(pow (+ (* a a) (* b b)) 2)
2.0ms
(* 4 (* b b))
0.0ms
(expm1 (log1p (pow (+ (* a a) (* b b)) 2)))

series219.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
85.0ms
(expm1 (log1p (pow (+ (* a a) (* b b)) 2)))
82.0ms
(+ (expm1 (log1p (pow (+ (* a a) (* b b)) 2))) (* 4 (* b b)))
33.0ms
(pow (+ (* a a) (* b b)) 2)
20.0ms
(* 4 (* b b))

simplify2.2s

Counts
22 → 67
Calls
22 calls:
Slowest
436.0ms
(- (+ (* 2 (/ (* (exp (- (* 4 (log (/ 1 a))))) (pow b 2)) (pow a 2))) (+ (* 4 (pow b 2)) (exp (- (* 4 (log (/ 1 a))))))) 1)
414.0ms
(- (+ (* 2 (/ (* (exp (- (* 4 (log (/ -1 a))))) (pow b 2)) (pow a 2))) (+ (* 4 (pow b 2)) (exp (- (* 4 (log (/ -1 a))))))) 1)
202.0ms
(* (exp (expm1 (log1p (pow (+ (* a a) (* b b)) 2)))) (exp (* 4 (* b b))))
167.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
151.0ms
(* (cbrt (expm1 (log1p (pow (+ (* a a) (* b b)) 2)))) (cbrt (expm1 (log1p (pow (+ (* a a) (* b b)) 2)))))

prune733.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes112.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample3.9s

Algorithm
intervals
Results
20583×(pre true 80)
20583×(body real 80)