Average Error: 58.1 → 0.0
Time: 1.2m
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 2018362 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 56.2s)Debug log

start375.0ms

Algorithm
intervals

setup54.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize24.0ms

Local error

Found 3 expressions with local error:

7.0b
(- (exp x) (exp (- x)))
1.0b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

14.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
6.0ms
(- (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))

series188.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

100.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
55.0ms
(- (exp x) (exp (- x)))
33.0ms
(+ (exp x) (exp (- x)))

simplify1.9s

Counts
69 → 82
Calls

69 calls. Slowest were:

460.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
314.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
162.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))

prune550.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

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)

series96.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

96.0ms
(tanh x)

simplify111.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

62.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
26.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
21.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune111.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize20.0ms

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)))

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))

series68.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

29.0ms
(* 2/15 (pow x 5))
24.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
16.0ms
(* 1/3 (pow x 3))

simplify427.0ms

Counts
17 → 46
Calls

17 calls. Slowest were:

140.0ms
(/ (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)))
73.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune668.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

38.7b
(cbrt (* (* (tanh x) (tanh x)) (tanh x)))
0.1b
(* (* (tanh x) (tanh x)) (tanh x))
0.0b
(tanh x)
0.0b
(tanh x)

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 181
Calls

4 calls. Slowest were:

10.0ms
(* (* (tanh x) (tanh x)) (tanh x))
7.0ms
(cbrt (* (* (tanh x) (tanh x)) (tanh x)))
0.0ms
(tanh x)

series391.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

109.0ms
(* (* (tanh x) (tanh x)) (tanh x))
103.0ms
(tanh x)
93.0ms
(cbrt (* (* (tanh x) (tanh x)) (tanh x)))
86.0ms
(tanh x)

simplify49.2s

Counts
204 → 193
Calls

204 calls. Slowest were:

987.0ms
(* (* (- 1 (exp (* -2 x))) (- (exp (* 2 x)) 1)) (- 1 (exp (* -2 x))))
787.0ms
(* (+ (exp x) (exp (- x))) (+ (exp (* 2 x)) 1))
723.0ms
(cbrt (* (* (- (exp (* 2 x)) 1) (- 1 (exp (* -2 x)))) (- 1 (exp (* -2 x)))))

prune2.0s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes19.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms