Average Error: 0.2 → 0.0
Time: 14.4s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot \left(1 - a\right)\right) \cdot a\right) + \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
  3. Using strategy rm

  4. Applied associate-*l*0.0

    \[\leadsto \left(\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right) + 4 \cdot \left(\color{blue}{a \cdot \left(a \cdot \left(1 - a\right)\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  5. Final simplification0.0

    \[\leadsto \left(4 \cdot \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot \left(1 - a\right)\right) \cdot a\right) + \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right)\right) - 1\]

Reproduce

herbie shell --seed 2019010 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))

Details

Time bar (total: 14.2s)Debug log

sample259.0ms

Algorithm
intervals

simplify122.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
122.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1)

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize39.0ms

Local error

Found 4 expressions with local error:

1.1b
(* (* b b) (+ 3 a))
0.3b
(pow (+ (* a a) (* b b)) 2)
0.1b
(* (* a a) (- 1 a))
0.0b
(* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
18×associate-*r/
9×add-exp-log
9×associate-*r*
9×pow1
8×add-cube-cbrt
8×add-cbrt-cube
8×*-un-lft-identity
8×add-sqr-sqrt
4×add-log-exp
4×flip-+
4×frac-add
4×flip3-+
4×distribute-lft-in
4×distribute-rgt-in
3×flip--
3×*-commutative
3×flip3--
3×unpow-prod-down
2×cbrt-unprod
2×prod-exp
2×pow-prod-down
2×associate-*l*
2×sub-neg
1×unpow2
1×pow-exp
1×pow-pow
1×pow-to-exp
Counts
4 → 73
Calls
4 calls:
Slowest
10.0ms
(* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))
7.0ms
(pow (+ (* a a) (* b b)) 2)
3.0ms
(* (* a a) (- 1 a))
3.0ms
(* (* b b) (+ 3 a))

series138.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))
40.0ms
(* (* a a) (- 1 a))
30.0ms
(pow (+ (* a a) (* b b)) 2)
18.0ms
(* (* b b) (+ 3 a))

simplify3.4s

Counts
50 → 85
Calls
50 calls:
Slowest
452.0ms
(* (* (* (* a a) (* a a)) (* a a)) (* (* (- 1 a) (- 1 a)) (- 1 a)))
313.0ms
(- (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))
210.0ms
(* 4 (* (* b b) (+ 3 a)))
203.0ms
(+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))
201.0ms
(sqrt (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))

prune1.1s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize31.0ms

Local error

Found 4 expressions with local error:

3.4b
(* (pow a 2) (pow b 2))
1.1b
(* (* b b) (+ 3 a))
0.1b
(* (* a a) (- 1 a))
0.0b
(* 2 (* (pow a 2) (pow b 2)))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
12×add-exp-log
11×associate-*r*
10×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×pow1
8×add-sqr-sqrt
6×prod-exp
6×associate-*l*
4×add-log-exp
4×*-commutative
4×associate-*r/
4×pow-to-exp
3×cbrt-unprod
3×pow-prod-down
3×distribute-lft-in
3×distribute-rgt-in
2×unpow2
2×sub-neg
1×flip--
1×flip-+
1×flip3--
1×flip3-+
Counts
4 → 71
Calls
4 calls:
Slowest
5.0ms
(* 2 (* (pow a 2) (pow b 2)))
4.0ms
(* (* a a) (- 1 a))
3.0ms
(* (pow a 2) (pow b 2))
3.0ms
(* (* b b) (+ 3 a))

series96.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
26.0ms
(* (pow a 2) (pow b 2))
26.0ms
(* (* a a) (- 1 a))
23.0ms
(* 2 (* (pow a 2) (pow b 2)))
20.0ms
(* (* b b) (+ 3 a))

simplify1.2s

Counts
46 → 83
Calls
46 calls:
Slowest
417.0ms
(* (* (* (* a a) (* a a)) (* a a)) (* (* (- 1 a) (- 1 a)) (- 1 a)))
122.0ms
(* (* (* (* b b) (* b b)) (* b b)) (* (* (+ 3 a) (+ 3 a)) (+ 3 a)))
112.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))
68.0ms
(* 2 (* (pow a 2) (pow b 2)))
64.0ms
(* 2 (* (pow a 2) (pow b 2)))

prune1.3s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

3.4b
(* (pow a 2) (pow b 2))
1.1b
(* (* b b) (+ 3 a))
0.1b
(* a (* a (- 1 a)))
0.0b
(* 2 (* (pow a 2) (pow b 2)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
9×associate-*r*
9×distribute-lft-in
9×distribute-rgt-in
8×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*r/
6×pow1
5×prod-exp
5×associate-*l*
4×add-log-exp
4×*-commutative
4×sub-neg
4×pow-to-exp
2×unpow2
2×cbrt-unprod
2×pow-prod-down
1×flip--
1×flip-+
1×flip3--
1×flip3-+
Counts
4 → 69
Calls
4 calls:
Slowest
6.0ms
(* (pow a 2) (pow b 2))
6.0ms
(* (* b b) (+ 3 a))
4.0ms
(* 2 (* (pow a 2) (pow b 2)))
2.0ms
(* a (* a (- 1 a)))

series112.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
36.0ms
(* a (* a (- 1 a)))
31.0ms
(* (* b b) (+ 3 a))
29.0ms
(* 2 (* (pow a 2) (pow b 2)))
16.0ms
(* (pow a 2) (pow b 2))

simplify852.0ms

Counts
45 → 81
Calls
45 calls:
Slowest
145.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))
115.0ms
(* (* (* (* b b) (* b b)) (* b b)) (* (* (+ 3 a) (+ 3 a)) (+ 3 a)))
62.0ms
(+ (* 3 3) (- (* a a) (* 3 a)))
52.0ms
(* 2 (* (pow a 2) (pow b 2)))
45.0ms
(* 2 (* (pow a 2) (pow b 2)))

prune1.2s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample4.2s

Algorithm
intervals