Average Error: 0.0 → 0.0
Time: 10.2s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{2}{e^{x} + e^{-x}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}}\]

  2. Final simplification0.0

    \[\leadsto \frac{2}{e^{x} + e^{-x}}\]

Reproduce

herbie shell --seed 2019016 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 9.9s)Debug log

sample22.0ms

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

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(/ 2 (+ (exp x) (exp (- x))))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ 2 (+ (exp x) (exp (- x))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
4×*-un-lft-identity
4×associate-/r*
2×add-cube-cbrt
2×associate-/r/
2×add-sqr-sqrt
1×add-log-exp
1×distribute-lft-out
1×div-inv
1×flip-+
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×frac-2neg
1×clear-num
1×pow1
Counts
1 → 16
Calls
1 calls:
Slowest
5.0ms
(/ 2 (+ (exp x) (exp (- x))))

series51.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
51.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify244.0ms

Counts
9 → 19
Calls
9 calls:
Slowest
163.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
44.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
15.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))
4.0ms
(- (exp x) (exp (- x)))
4.0ms
(/ 2 (+ (exp x) (exp (- x))))

prune178.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize36.0ms

Local error

Found 3 expressions with local error:

12.4b
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
0.1b
(* 5/24 (pow x 4))
0.0b
(* 1/2 (pow x 2))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
7×associate-*r*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×add-exp-log
3×add-cbrt-cube
3×pow1
2×*-commutative
1×unpow2
1×flip--
1×associate--l+
1×diff-log
1×flip3--
1×sub-neg
Counts
3 → 36
Calls
3 calls:
Slowest
23.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
2.0ms
(* 5/24 (pow x 4))
2.0ms
(* 1/2 (pow x 2))

series80.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
31.0ms
(* 5/24 (pow x 4))
29.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
19.0ms
(* 1/2 (pow x 2))

simplify287.0ms

Counts
17 → 45
Calls
17 calls:
Slowest
77.0ms
(/ (exp (+ (* 5/24 (pow x 4)) 1)) (exp (* 1/2 (pow x 2))))
70.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
66.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
35.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
7.0ms
(* 1/2 (pow x 2))

prune391.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

3.7b
(- (exp x) (exp (- x)))
3.7b
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
0.5b
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp x) (exp (- x))))
0.0b
(* (exp x) (exp x))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
12×add-sqr-sqrt
11×pow1
9×associate-*l*
8×add-log-exp
8×add-cube-cbrt
8×associate-*r*
8×add-cbrt-cube
6×add-exp-log
5×flip--
5×flip3--
4×sub-neg
3×difference-of-squares
2×distribute-lft-out--
2×cbrt-unprod
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×pow-prod-down
2×diff-log
2×distribute-lft-in
2×frac-times
2×distribute-rgt-in
1×div-inv
1×pow-prod-up
1×associate-*l/
1×pow-plus
1×pow2
Counts
4 → 78
Calls
4 calls:
Slowest
16.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp x) (exp (- x))))
5.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
4.0ms
(- (exp x) (exp (- x)))
3.0ms
(* (exp x) (exp x))

series244.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
141.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp x) (exp (- x))))
53.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
41.0ms
(- (exp x) (exp (- x)))
10.0ms
(* (exp x) (exp x))

simplify6.4s

Counts
53 → 90
Calls
53 calls:
Slowest
537.0ms
(* (- (exp (- x))) (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
510.0ms
(* (- (exp (- x))) (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
490.0ms
(* 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
487.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp (- x))))
443.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp (- x))))

prune858.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes80.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample878.0ms

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)