Average Error: 58.0 → 0.0
Time: 21.3s
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 58.0

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2018365 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 4.8s)Debug log

start475.0ms

Algorithm
intervals

setup60.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize29.0ms

Local error

Found 3 expressions with local error:

7.8b
(- (exp x) (exp (- x)))
0.8b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

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

series230.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

121.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
66.0ms
(- (exp x) (exp (- x)))
43.0ms
(+ (exp x) (exp (- x)))

simplify1.9s

Counts
69 → 82
Calls

69 calls. Slowest were:

454.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
274.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
167.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))

prune578.0ms

Pruning

2 alts after pruning (2 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)

series81.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

81.0ms
(tanh x)

simplify120.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

58.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
36.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
26.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))

prune81.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 3 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

6.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
2.0ms
(* 2/15 (pow x 5))
1.0ms
(* 1/3 (pow x 3))

series75.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

35.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
22.0ms
(* 2/15 (pow x 5))
18.0ms
(* 1/3 (pow x 3))

simplify405.0ms

Counts
17 → 46
Calls

17 calls. Slowest were:

126.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
89.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
67.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune616.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes31.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms