Average Error: 0.2 → 0.0
Time: 15.8s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(1 - a\right) \cdot \left(a \cdot a\right)\right) + (2 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right) - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  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. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))

Details

Time bar (total: 15.4s)Debug log

sample159.0ms

Algorithm
intervals
Results
140.0ms661×body80valid

simplify173.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
173.0ms
(- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1)

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize41.0ms

Local error

Found 4 expressions with local error:

0.8b
(* (* b b) (+ 3 a))
0.4b
(pow (+ (* a a) (* b b)) 2)
0.1b
(* (* a a) (- 1 a))
0.0b
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×add-exp-log
9×*-un-lft-identity
9×pow1
9×add-sqr-sqrt
8×add-cbrt-cube
7×fma-def
6×add-log-exp
6×associate-*r*
6×unpow-prod-down
4×log1p-expm1-u
4×associate-*r/
4×distribute-lft-in
4×expm1-log1p-u
4×distribute-rgt-in
2×unpow2
2×flip-+
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×pow-prod-down
2×associate-*l*
2×flip3-+
2×sub-neg
2×associate-+r+
1×flip--
1×pow-exp
1×flip3--
1×pow-pow
1×sum-log
1×+-commutative
1×pow-to-exp
Counts
4 → 82
Calls
4 calls:
Slowest
20.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))
3.0ms
(* (* a a) (- 1 a))
3.0ms
(* (* b b) (+ 3 a))
2.0ms
(pow (+ (* a a) (* b b)) 2)

series132.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
64.0ms
(+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))
26.0ms
(* (* b b) (+ 3 a))
23.0ms
(* (* a a) (- 1 a))
20.0ms
(pow (+ (* a a) (* b b)) 2)

simplify2.2s

Counts
48 → 94
Calls
48 calls:
Slowest
345.0ms
(* (* (* (* a a) (* a a)) (* a a)) (* (* (- 1 a) (- 1 a)) (- 1 a)))
262.0ms
(* (* (* b b) (+ 3 a)) 4)
256.0ms
(* 4 (* (* b b) (+ 3 a)))
224.0ms
(- (+ (* 4 (pow a 2)) (* 12 (pow b 2))) (* 4 (pow a 3)))
112.0ms
(* (exp (pow (+ (* a a) (* b b)) 2)) (exp (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))))

prune979.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 4 expressions with local error:

0.8b
(* (* b b) (+ 3 a))
0.1b
(* (* a b) (* a b))
0.1b
(* (* a a) (- 1 a))
0.0b
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
13×pow1
10×add-exp-log
10×add-cbrt-cube
7×add-cube-cbrt
7×associate-*r*
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-log-exp
4×log1p-expm1-u
4×associate-*r/
4×distribute-lft-in
4×expm1-log1p-u
4×distribute-rgt-in
3×cbrt-unprod
3×*-commutative
3×prod-exp
3×pow-prod-down
3×associate-*l*
3×fma-def
2×flip-+
2×flip3-+
2×sub-neg
2×associate-+r+
1×flip--
1×fma-udef
1×pow-prod-up
1×flip3--
1×pow-plus
1×sum-log
1×associate-+l+
1×+-commutative
1×pow2
Counts
4 → 81
Calls
4 calls:
Slowest
4.0ms
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))
4.0ms
(* (* b b) (+ 3 a))
4.0ms
(* (* a a) (- 1 a))
3.0ms
(* (* a b) (* a b))

series132.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
68.0ms
(+ (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))
25.0ms
(* (* a a) (- 1 a))
23.0ms
(* (* b b) (+ 3 a))
16.0ms
(* (* a b) (* a b))

simplify2.6s

Counts
46 → 93
Calls
46 calls:
Slowest
495.0ms
(* (* (* (* a b) (* a b)) (* a b)) (* (* (* a b) (* a b)) (* a b)))
385.0ms
(* (* (* (* a a) (* a a)) (* a a)) (* (* (- 1 a) (- 1 a)) (- 1 a)))
273.0ms
(* (* (* b b) (+ 3 a)) 4)
248.0ms
(- (+ (* 4 (pow a 2)) (* 12 (pow b 2))) (* 4 (pow a 3)))
247.0ms
(* 4 (* (* b b) (+ 3 a)))

prune971.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

9.3b
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
9.3b
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
0.8b
(* (* b b) (+ 3 a))
0.2b
(fma (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×sqrt-prod
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*r*
2×associate-*r/
2×pow1/2
2×rem-sqrt-square
1×fma-udef
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×flip3-+
1×distribute-lft-in
1×distribute-rgt-in
Counts
4 → 59
Calls
4 calls:
Slowest
4.0ms
(* (* b b) (+ 3 a))
1.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
1.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
0.0ms
(fma (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))

series185.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
94.0ms
(fma (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a)))))
37.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
36.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
18.0ms
(* (* b b) (+ 3 a))

simplify948.0ms

Counts
30 → 71
Calls
30 calls:
Slowest
195.0ms
(- (+ (* 4 (pow a 2)) (* 12 (pow b 2))) (* 4 (pow a 3)))
86.0ms
(* (* (* (* b b) (* b b)) (* b b)) (* (* (+ 3 a) (+ 3 a)) (+ 3 a)))
64.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
64.0ms
(+ (pow b 4) (+ (pow a 4) (* 2 (* (pow a 2) (pow b 2)))))
63.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))

prune835.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

10.6b
(sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))
10.6b
(sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))
9.3b
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
9.3b
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
26×sqrt-prod
16×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
6×rem-sqrt-square
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×pow1/2
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
Counts
4 → 64
Calls
4 calls:
Slowest
1.0ms
(sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))
1.0ms
(sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))
1.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
1.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))

series194.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
63.0ms
(sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))
59.0ms
(sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))
37.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))
36.0ms
(sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))

simplify1.1s

Counts
36 → 76
Calls
36 calls:
Slowest
63.0ms
(sqrt (* (cbrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))) (cbrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))))
63.0ms
(sqrt (* (cbrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))) (cbrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))))
63.0ms
(sqrt (sqrt (* (cbrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))) (cbrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4)))))))
63.0ms
(sqrt (sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))))
62.0ms
(sqrt (sqrt (sqrt (fma 2 (* (* a b) (* a b)) (+ (pow a 4) (pow b 4))))))

prune1.2s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

regimes78.0ms

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
Results
3.0s20530×body80valid