Average Error: 58.1 → 0.0
Time: 20.7s
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 58.1

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019007 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 20.2s)Debug log

sample328.0ms

Algorithm
intervals

simplify21.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

21.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize28.0ms

Local error

Found 3 expressions with local error:

7.5b
(- (exp x) (exp (- x)))
1.2b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite63.0ms

Algorithm
rewrite-expression-head
Counts
3 → 94
Calls

3 calls. Slowest were:

47.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
10.0ms
(- (exp x) (exp (- x)))
5.0ms
(+ (exp x) (exp (- x)))

series171.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

97.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
42.0ms
(- (exp x) (exp (- x)))
32.0ms
(+ (exp x) (exp (- x)))

simplify2.4s

Counts
91 → 103
Calls

91 calls. Slowest were:

354.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
249.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
140.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))

prune829.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize5.0ms

Local error

Found 1 expressions with local error:

0.0b
(tanh x)

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

0.0ms
(tanh x)

series86.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

86.0ms
(tanh x)

simplify187.0ms

Counts
3 → 15
Calls

3 calls. Slowest were:

107.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
51.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
28.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune90.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 3 expressions with local error:

0.1b
(* 1/3 (pow x 3))
0.1b
(* 2/15 (pow x 5))
0.0b
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 49
Calls

3 calls. Slowest were:

6.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
3.0ms
(* 1/3 (pow x 3))
2.0ms
(* 2/15 (pow x 5))

series68.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

30.0ms
(* 2/15 (pow x 5))
23.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
15.0ms
(* 1/3 (pow x 3))

simplify921.0ms

Counts
25 → 58
Calls

25 calls. Slowest were:

140.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
131.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
124.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))

prune493.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize28.0ms

Local error

Found 3 expressions with local error:

7.5b
(- (exp (* 2 x)) 1)
0.7b
(/ (- (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1))
0.0b
(+ (exp (* 2 x)) 1)

rewrite48.0ms

Algorithm
rewrite-expression-head
Counts
3 → 71
Calls

3 calls. Slowest were:

28.0ms
(/ (- (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1))
11.0ms
(- (exp (* 2 x)) 1)
8.0ms
(+ (exp (* 2 x)) 1)

series82.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

59.0ms
(/ (- (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1))
16.0ms
(- (exp (* 2 x)) 1)
8.0ms
(+ (exp (* 2 x)) 1)

simplify2.6s

Counts
52 → 80
Calls

52 calls. Slowest were:

346.0ms
(/ (* (* (- (exp (* 2 x)) 1) (- (exp (* 2 x)) 1)) (- (exp (* 2 x)) 1)) (* (* (+ (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1)) (+ (exp (* 2 x)) 1)))
171.0ms
(+ (* 2 x) (+ (* 4/3 (pow x 3)) (* 2 (pow x 2))))
141.0ms
(- (* (exp (* 2 x)) (exp (* 2 x))) (* 1 1))

prune597.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes36.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample11.1s

Algorithm
intervals