Average Error: 57.9 → 0.0
Time: 13.1s
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 57.9

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019008 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 12.9s)Debug log

sample280.0ms

Algorithm
intervals

simplify37.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

37.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize26.0ms

Local error

Found 3 expressions with local error:

4.9b
(- (exp x) (exp (- x)))
0.7b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite61.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

43.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
8.0ms
(+ (exp x) (exp (- x)))
8.0ms
(- (exp x) (exp (- x)))

series193.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

113.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
50.0ms
(- (exp x) (exp (- x)))
31.0ms
(+ (exp x) (exp (- x)))

simplify1.7s

Counts
69 → 82
Calls

69 calls. Slowest were:

332.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
252.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
190.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))

prune486.0ms

Pruning

2 alts after pruning (2 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 → 10
Calls

1 calls. Slowest were:

0.0ms
(tanh x)

series78.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

78.0ms
(tanh x)

simplify86.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

43.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
22.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
21.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune57.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

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:

5.0ms
(- (+ 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))

series63.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

23.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
20.0ms
(* 2/15 (pow x 5))
19.0ms
(* 1/3 (pow x 3))

simplify279.0ms

Counts
17 → 46
Calls

17 calls. Slowest were:

84.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
78.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
45.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune347.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes17.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample9.2s

Algorithm
intervals