Average Error: 0.0 → 0.0
Time: 4.7s
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 2019022 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 4.4s)Debug log

sample18.0ms

Algorithm
intervals
Results
12.0ms256×body80valid

simplify5.0ms

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
4×add-log-exp
4×add-cube-cbrt
4×associate-/r*
4×add-sqr-sqrt
3×fma-def
2×distribute-lft-out
2×log1p-expm1-u
2×flip-+
2×associate-/r/
2×add-exp-log
2×add-cbrt-cube
2×flip3-+
2×pow1
2×expm1-log1p-u
1×div-inv
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
Counts
2 → 35
Calls
2 calls:
Slowest
4.0ms
(+ (exp x) (exp (- x)))
3.0ms
(/ 2 (+ (exp x) (exp (- x))))

series65.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
38.0ms
(/ 2 (+ (exp x) (exp (- x))))
27.0ms
(+ (exp x) (exp (- x)))

simplify418.0ms

Counts
16 → 41
Calls
16 calls:
Slowest
246.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
88.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
14.0ms
(+ (exp (* -1 x)) (exp x))
11.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) 2))
9.0ms
(* (exp (exp x)) (exp (exp (- x))))

prune333.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.1b
(log (exp (/ 2 (+ (exp x) (exp (- x))))))
0.0b
(/ 2 (+ (exp x) (exp (- x))))
0.0b
(exp (/ 2 (+ (exp x) (exp (- x)))))
0.0b
(+ (exp x) (exp (- x)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
13×*-un-lft-identity
12×exp-prod
9×add-cube-cbrt
9×add-sqr-sqrt
7×add-log-exp
7×log-pow
6×associate-/r/
5×pow1
4×log1p-expm1-u
4×flip-+
4×add-exp-log
4×add-cbrt-cube
4×flip3-+
4×associate-/r*
4×expm1-log1p-u
3×div-inv
3×fma-def
3×log-prod
2×distribute-lft-out
1×rem-exp-log
1×rem-log-exp
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
Counts
4 → 71
Calls
4 calls:
Slowest
4.0ms
(+ (exp x) (exp (- x)))
4.0ms
(log (exp (/ 2 (+ (exp x) (exp (- x))))))
3.0ms
(/ 2 (+ (exp x) (exp (- x))))
2.0ms
(exp (/ 2 (+ (exp x) (exp (- x)))))

series210.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
89.0ms
(exp (/ 2 (+ (exp x) (exp (- x)))))
53.0ms
(/ 2 (+ (exp x) (exp (- x))))
43.0ms
(log (exp (/ 2 (+ (exp x) (exp (- x))))))
25.0ms
(+ (exp x) (exp (- x)))

simplify1.6s

Counts
38 → 83
Calls
38 calls:
Slowest
345.0ms
(exp (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
281.0ms
(- (+ (* 1/3 (* (pow x 4) E)) E) (* 1/2 (* (pow x 2) E)))
269.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
242.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
107.0ms
(exp (/ 2 (+ (pow (exp x) 3) (pow (exp (- x)) 3))))

prune778.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes86.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample844.0ms

Algorithm
intervals
Results
739.0ms8000×body80valid