Average Error: 0.0 → 0.0
Time: 3.1s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}}\]

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 2.9s)Debug log

sample31.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(/ 2 (+ (exp x) (exp (- x))))

prune5.0ms

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))))

series60.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
60.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify341.0ms

Counts
9 → 21
Calls
9 calls:
Slowest
207.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
86.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
14.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))
7.0ms
(/ 2 (+ (exp x) (exp (- x))))
7.0ms
(cbrt (+ (exp x) (exp (- x))))

prune204.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
8×associate-/r*
4×add-cube-cbrt
4×associate-/r/
4×add-sqr-sqrt
2×add-log-exp
2×distribute-lft-out
2×div-inv
2×log1p-expm1-u
2×flip-+
2×add-exp-log
2×add-cbrt-cube
2×flip3-+
2×frac-2neg
2×clear-num
2×pow1
2×expm1-log1p-u
Counts
2 → 36
Calls
2 calls:
Slowest
6.0ms
(/ 2 (+ (exp x) (exp (- x))))
6.0ms
(/ 2 (+ (exp x) (exp (- x))))

series117.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
60.0ms
(/ 2 (+ (exp x) (exp (- x))))
57.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify785.0ms

Counts
18 → 42
Calls
18 calls:
Slowest
279.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
258.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
64.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
58.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
17.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))

prune540.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample783.0ms

Algorithm
intervals