Average Error: 58.0 → 0.0
Time: 21.2s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 20.9s)Debug log

sample295.0ms

Algorithm
intervals

simplify28.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

28.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize14.0ms

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)))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
3 → 94
Calls

3 calls. Slowest were:

23.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
5.0ms
(- (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))

series170.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
57.0ms
(- (exp x) (exp (- x)))
26.0ms
(+ (exp x) (exp (- x)))

simplify2.4s

Counts
91 → 103
Calls

91 calls. Slowest were:

345.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
262.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
190.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))

prune765.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize2.0ms

Local error

Found 1 expressions with local error:

0.0b
(tanh x)

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

0.0ms
(tanh x)

series100.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

100.0ms
(tanh x)

simplify211.0ms

Counts
3 → 15
Calls

3 calls. Slowest were:

128.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
46.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
36.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune154.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

12.0ms
(/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) (+ (exp x) (exp (- x))))
3.0ms
(* (pow x 5) 1/60)
0.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))

series155.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) (+ (exp x) (exp (- x))))
37.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
34.0ms
(fma 1/3 (* x x) 2)
28.0ms
(* (pow x 5) 1/60)

simplify2.4s

Counts
48 → 80
Calls

48 calls. Slowest were:

229.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
141.0ms
(+ (* 1/3 (/ (pow x 3) (+ (exp x) (exp (- x))))) (+ (* 2 (/ x (+ (exp x) (exp (- x))))) (* 1/60 (/ (pow x 5) (+ (exp x) (exp (- x)))))))
135.0ms
(+ (* 1/3 (/ (pow x 3) (+ (exp (* -1 x)) (exp x)))) (+ (* 2 (/ x (+ (exp (* -1 x)) (exp x)))) (* 1/60 (/ (pow x 5) (+ (exp (* -1 x)) (exp x))))))

prune1.0s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize20.0ms

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)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

5.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
1.0ms
(* 1/3 (pow x 3))
1.0ms
(+ x (* 2/15 (pow x 5)))

series159.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

73.0ms
(+ x (* 2/15 (pow x 5)))
34.0ms
(* 2/15 (pow x 5))
32.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
20.0ms
(* 1/3 (pow x 3))

simplify1.0s

Counts
28 → 73
Calls

28 calls. Slowest were:

171.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))
165.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
120.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune648.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes25.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample11.2s

Algorithm
intervals