Average Error: 58.0 → 0.0
Time: 19.1s
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 58.0

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019005 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 18.6s)Debug log

sample285.0ms

Algorithm
intervals

simplify23.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize29.0ms

Local error

Found 3 expressions with local error:

6.5b
(- (exp x) (exp (- x)))
1.5b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

27.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
10.0ms
(+ (exp x) (exp (- x)))
7.0ms
(- (exp x) (exp (- x)))

series179.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

115.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
40.0ms
(- (exp x) (exp (- x)))
24.0ms
(+ (exp x) (exp (- x)))

simplify1.8s

Counts
69 → 82
Calls

69 calls. Slowest were:

490.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
230.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
155.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))

prune576.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize3.0ms

Local error

Found 1 expressions with local error:

0.0b
(tanh x)

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(tanh x)

series72.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

72.0ms
(tanh x)

simplify104.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

58.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
26.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
19.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune93.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize27.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 1/3 (pow x 3))
0.1b
(* 1/60 (pow x 5))
0.0b
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
0.0b
(/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) (+ (exp x) (exp (- x))))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

35.0ms
(/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) (+ (exp x) (exp (- x))))
6.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
2.0ms
(* 1/3 (pow x 3))

series207.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

86.0ms
(/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) (+ (exp x) (exp (- x))))
83.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
21.0ms
(* 1/60 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify2.1s

Counts
56 → 82
Calls

56 calls. Slowest were:

266.0ms
(- (* (* 2 x) (* 2 x)) (* (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))))
185.0ms
(/ (* (* (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))) (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
146.0ms
(+ (pow (* 2 x) 3) (pow (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))) 3))

prune856.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 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)))
0.0b
(+ x (* 2/15 (pow x 5)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

4.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
1.0ms
(* 1/3 (pow x 3))
1.0ms
(* 2/15 (pow x 5))

series138.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(+ x (* 2/15 (pow x 5)))
27.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
27.0ms
(* 2/15 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify407.0ms

Counts
20 → 59
Calls

20 calls. Slowest were:

165.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
73.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
54.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune535.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes53.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample11.0s

Algorithm
intervals