Average Error: 0.2 → 0.0
Time: 18.5s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right)\right) - 1\]

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

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Details

Time bar (total: 18.0s)Debug log

sample144.0ms

Algorithm
intervals

simplify364.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
364.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1)

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize35.0ms

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))

rewrite37.0ms

Algorithm
rewrite-expression-head
Rules
8×add-log-exp
5×add-cube-cbrt
5×add-exp-log
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×add-cbrt-cube
3×unpow-prod-down
2×flip-+
2×flip3-+
2×sum-log
2×+-commutative
1×unpow2
1×*-commutative
1×pow-exp
1×associate-*r*
1×pow-pow
1×pow-to-exp
Counts
4 → 45
Calls
4 calls:
Slowest
25.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
4.0ms
(+ (* a a) (* b b))
4.0ms
(pow (+ (* a a) (* b b)) 2)
1.0ms
(* 4 (* b b))

series139.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
57.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b)))
37.0ms
(pow (+ (* a a) (* b b)) 2)
24.0ms
(+ (* a a) (* b b))
20.0ms
(* 4 (* b b))

simplify786.0ms

Counts
19 → 57
Calls
19 calls:
Slowest
121.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))
118.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
115.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (* b b))))
113.0ms
(+ (* 2 (* (pow a 2) (pow b 2))) (+ (* 4 (pow b 2)) (pow a 4)))
90.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))

prune684.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize55.0ms

Local error

Found 4 expressions with local error:

3.4b
(* (pow a 2) (pow b 2))
0.0b
(* 4 (* b b))
0.0b
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
0.0b
(* 2 (* (pow a 2) (pow b 2)))

rewrite70.0ms

Algorithm
rewrite-expression-head
Rules
13×add-log-exp
8×add-exp-log
6×add-cube-cbrt
6×associate-*r*
6×add-cbrt-cube
6×*-un-lft-identity
6×sum-log
6×add-sqr-sqrt
4×prod-exp
4×associate-*l*
4×pow1
4×pow-to-exp
3×*-commutative
2×unpow2
1×flip-+
1×cbrt-unprod
1×pow-prod-down
1×flip3-+
1×associate-+l+
1×+-commutative
Counts
4 → 54
Calls
4 calls:
Slowest
53.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
8.0ms
(* 2 (* (pow a 2) (pow b 2)))
7.0ms
(* (pow a 2) (pow b 2))
2.0ms
(* 4 (* b b))

series108.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
43.0ms
(+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))
28.0ms
(* 2 (* (pow a 2) (pow b 2)))
25.0ms
(* (pow a 2) (pow b 2))
11.0ms
(* 4 (* b b))

simplify1.4s

Counts
28 → 66
Calls
28 calls:
Slowest
341.0ms
(* (* (exp (pow b 4)) (* (exp (* 2 (* (pow a 2) (pow b 2)))) (exp (pow a 4)))) (exp (* 4 (* b b))))
222.0ms
(* (* (exp (pow b 4)) (exp (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))
202.0ms
(* (exp (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4)))) (exp (* 4 (* b b))))
124.0ms
(* (* (* (pow a 2) (pow a 2)) (pow a 2)) (* (* (pow b 2) (pow b 2)) (pow b 2)))
117.0ms
(+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (* 4 (pow b 2))))

prune938.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

14.2b
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
14.2b
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
3.4b
(* (pow a 2) (pow b 2))
3.4b
(* (pow a 2) (pow b 2))

rewrite134.0ms

Algorithm
rewrite-expression-head
Rules
12×add-exp-log
12×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
8×prod-exp
8×associate-*l*
8×associate-*r*
8×add-cbrt-cube
8×pow-to-exp
6×sqrt-prod
4×add-log-exp
4×unpow2
4×sqrt-div
4×pow1
2×flip-+
2×cbrt-unprod
2×*-commutative
2×pow-prod-down
2×pow1/2
2×flip3-+
2×rem-sqrt-square
Counts
4 → 72
Calls
4 calls:
Slowest
74.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
51.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
4.0ms
(* (pow a 2) (pow b 2))
3.0ms
(* (pow a 2) (pow b 2))

series228.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
95.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
84.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
31.0ms
(* (pow a 2) (pow b 2))
18.0ms
(* (pow a 2) (pow b 2))

simplify2.9s

Counts
46 → 84
Calls
46 calls:
Slowest
373.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))
361.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))
353.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))
305.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))
159.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))

prune1.6s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize54.0ms

Local error

Found 4 expressions with local error:

14.9b
(cbrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
14.9b
(cbrt (+ (* 2 (* (* a b) (* a b))) (+ (+ (pow a 4) (pow b 4)) (* b (* b 4)))))
14.2b
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
3.4b
(* (pow a 2) (pow b 2))

rewrite90.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
8×add-exp-log
6×add-cbrt-cube
6×cbrt-prod
4×add-log-exp
4×prod-exp
4×associate-*l*
4×associate-*r*
4×cbrt-div
4×pow1
4×pow-to-exp
3×flip-+
3×sqrt-prod
3×flip3-+
2×unpow2
2×sqrt-div
2×pow1/3
1×cbrt-unprod
1×*-commutative
1×pow-prod-down
1×pow1/2
1×rem-sqrt-square
Counts
4 → 62
Calls
4 calls:
Slowest
40.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
26.0ms
(cbrt (+ (* 2 (* (* a b) (* a b))) (+ (+ (pow a 4) (pow b 4)) (* b (* b 4)))))
20.0ms
(cbrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
3.0ms
(* (pow a 2) (pow b 2))

series475.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
195.0ms
(cbrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
167.0ms
(cbrt (+ (* 2 (* (* a b) (* a b))) (+ (+ (pow a 4) (pow b 4)) (* b (* b 4)))))
87.0ms
(sqrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))
26.0ms
(* (pow a 2) (pow b 2))

simplify2.7s

Counts
39 → 74
Calls
39 calls:
Slowest
367.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))
329.0ms
(- (+ (* 1/12 (* (pow b 2) (exp (* 1/3 (+ (log 4) (* 2 (log b))))))) (exp (* 1/3 (+ (log 4) (* 2 (log b)))))) (* 1/24 (* (pow a 4) (exp (* 1/3 (+ (log 4) (* 2 (log b))))))))
319.0ms
(- (+ (pow a 2) (pow b 2)) (* 2 (/ (pow a 2) (pow b 2))))
171.0ms
(cbrt (* (cbrt (+ (* 2 (* (* a b) (* a b))) (+ (+ (pow a 4) (pow b 4)) (* b (* b 4))))) (cbrt (+ (* 2 (* (* a b) (* a b))) (+ (+ (pow a 4) (pow b 4)) (* b (* b 4)))))))
158.0ms
(sqrt (* (cbrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b)))) (cbrt (+ (+ (pow b 4) (+ (* 2 (* (pow a 2) (pow b 2))) (pow a 4))) (* 4 (* b b))))))

prune1.8s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes91.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample3.1s

Algorithm
intervals