Average Error: 58.0 → 0.0
Time: 23.0s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 6.4s)Debug log

start512.0ms

Algorithm
intervals

setup51.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize31.0ms

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)))

rewrite54.0ms

Algorithm
rewrite-expression-head
Counts
3 → 94
Calls

3 calls. Slowest were:

33.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
10.0ms
(- (exp x) (exp (- x)))
9.0ms
(+ (exp x) (exp (- x)))

series194.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

88.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
66.0ms
(- (exp x) (exp (- x)))
40.0ms
(+ (exp x) (exp (- x)))

simplify2.4s

Counts
91 → 103
Calls

91 calls. Slowest were:

367.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
307.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
172.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))

prune763.0ms

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)

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

0.0ms
(tanh x)

series81.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

80.0ms
(tanh x)

simplify185.0ms

Counts
3 → 15
Calls

3 calls. Slowest were:

112.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
50.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
22.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune119.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize26.0ms

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)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 49
Calls

3 calls. Slowest were:

10.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
3.0ms
(* 1/3 (pow x 3))
2.0ms
(* 2/15 (pow x 5))

series82.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

29.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
27.0ms
(* 1/3 (pow x 3))
26.0ms
(* 2/15 (pow x 5))

simplify1.0s

Counts
25 → 58
Calls

25 calls. Slowest were:

204.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
153.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))
133.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune808.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes37.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms