Average Error: 58.2 → 0.0
Time: 11.4s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\tanh x\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.2

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied tanh-undef0.0

    \[\leadsto \color{blue}{\tanh x}\]

  4. Final simplification0.0

    \[\leadsto \tanh x\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 11.3s)Debug log

sample215.0ms

Algorithm
intervals

simplify24.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

24.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.8b

localize14.0ms

Local error

Found 3 expressions with local error:

5.3b
(- (exp x) (exp (- x)))
0.8b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
3 → 94
Calls

3 calls. Slowest were:

22.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
5.0ms
(- (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))

series134.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

72.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
38.0ms
(- (exp x) (exp (- x)))
23.0ms
(+ (exp x) (exp (- x)))

simplify2.0s

Counts
91 → 103
Calls

91 calls. Slowest were:

323.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
284.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
161.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))

prune650.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize2.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune4.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample8.3s

Algorithm
intervals