Average Error: 58.1 → 0.0
Time: 17.3s
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 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 17.0s)Debug log

sample203.0ms

Algorithm
intervals

simplify21.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize26.0ms

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)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

15.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
4.0ms
(- (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))

series135.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

72.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
39.0ms
(- (exp x) (exp (- x)))
24.0ms
(+ (exp x) (exp (- x)))

simplify1.7s

Counts
69 → 82
Calls

69 calls. Slowest were:

362.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
200.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
147.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))

prune500.0ms

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)

series116.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

116.0ms
(tanh x)

simplify155.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

74.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
48.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
33.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune85.0ms

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)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

9.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
2.0ms
(* 1/3 (pow x 3))
2.0ms
(* 2/15 (pow x 5))

series76.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

32.0ms
(* 1/3 (pow x 3))
23.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
22.0ms
(* 2/15 (pow x 5))

simplify337.0ms

Counts
17 → 46
Calls

17 calls. Slowest were:

108.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
61.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
60.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune394.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize13.0ms

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)

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
3 → 58
Calls

3 calls. Slowest were:

13.0ms
(/ (- (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1))
4.0ms
(- (exp (* 2 x)) 1)
3.0ms
(+ (exp (* 2 x)) 1)

series94.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

69.0ms
(/ (- (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1))
18.0ms
(- (exp (* 2 x)) 1)
8.0ms
(+ (exp (* 2 x)) 1)

simplify2.2s

Counts
47 → 67
Calls

47 calls. Slowest were:

363.0ms
(/ (* (* (- (exp (* 2 x)) 1) (- (exp (* 2 x)) 1)) (- (exp (* 2 x)) 1)) (* (* (+ (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1)) (+ (exp (* 2 x)) 1)))
226.0ms
(+ (* (exp (* 2 x)) (exp (* 2 x))) (- (* 1 1) (* (exp (* 2 x)) 1)))
156.0ms
(- (* (exp (* 2 x)) (exp (* 2 x))) (* 1 1))

prune474.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes22.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample10.4s

Algorithm
intervals