Average Error: 58.0 → 0.0
Time: 13.2s
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 2019004 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 13.0s)Debug log

sample273.0ms

Algorithm
intervals

simplify32.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.1b

localize14.0ms

Local error

Found 3 expressions with local error:

4.9b
(- (exp x) (exp (- x)))
1.5b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

25.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
15.0ms
(- (exp x) (exp (- x)))
7.0ms
(+ (exp x) (exp (- x)))

series174.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

83.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
51.0ms
(- (exp x) (exp (- x)))
40.0ms
(+ (exp x) (exp (- x)))

simplify1.5s

Counts
69 → 82
Calls

69 calls. Slowest were:

403.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
250.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
149.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))

prune318.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize2.0ms

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)

series79.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

79.0ms
(tanh x)

simplify86.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

44.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
22.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
20.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune50.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize18.0ms

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)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 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))

series58.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

24.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
19.0ms
(* 2/15 (pow x 5))
15.0ms
(* 1/3 (pow x 3))

simplify326.0ms

Counts
17 → 46
Calls

17 calls. Slowest were:

80.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
80.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
79.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune287.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes15.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample9.7s

Algorithm
intervals