Average Error: 0.0 → 0.0
Time: 17.7s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 17.0s)Debug log

sample19.0ms

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

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(/ 2 (+ (exp x) (exp (- x))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

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×log1p-expm1-u
1×flip-+
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×frac-2neg
1×clear-num
1×pow1
1×expm1-log1p-u
Counts
1 → 18
Calls
1 calls:
Slowest
3.0ms
(/ 2 (+ (exp x) (exp (- x))))

series34.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
33.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify305.0ms

Counts
9 → 21
Calls
9 calls:
Slowest
220.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
50.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
10.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))
5.0ms
(- (exp x) (exp (- x)))
5.0ms
(/ 2 (+ (exp x) (exp (- x))))

prune156.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 1 expressions with local error:

12.4b
(fma 5/24 (pow x 4) (fma (* x x) -1/2 1))

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma 5/24 (pow x 4) (fma (* x x) -1/2 1))

series18.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
18.0ms
(fma 5/24 (pow x 4) (fma (* x x) -1/2 1))

simplify251.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
98.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
84.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
69.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))

prune142.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize31.0ms

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))

rewrite35.0ms

Algorithm
rewrite-expression-head
Rules
31×*-un-lft-identity
31×add-sqr-sqrt
28×prod-diff
27×add-cube-cbrt
11×distribute-lft-in
11×pow1
11×distribute-rgt-in
9×associate-*l*
8×add-log-exp
8×associate-*r*
8×add-cbrt-cube
6×add-exp-log
5×flip--
5×flip3--
4×log1p-expm1-u
4×fma-neg
4×sub-neg
4×expm1-log1p-u
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×frac-times
1×div-inv
1×pow-prod-up
1×associate-*l/
1×pow-plus
1×pow2
Counts
4 → 118
Calls
4 calls:
Slowest
15.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp x) (exp (- x))))
7.0ms
(* (exp x) (exp x))
6.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
5.0ms
(- (exp x) (exp (- x)))

series280.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
179.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp x) (exp (- x))))
51.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
39.0ms
(- (exp x) (exp (- x)))
10.0ms
(* (exp x) (exp x))

simplify13.4s

Counts
94 → 130
Calls
94 calls:
Slowest
754.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp (- x))))
753.0ms
(* 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
690.0ms
(* (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp (- x))))
625.0ms
(* (- (exp (- x))) (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
612.0ms
(* (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (* (exp x) (exp x)) (+ (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x))))))

prune1.3s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes111.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample837.0ms

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