Average Error: 58.1 → 0.0
Time: 1.3m
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 1.1m)Debug log

start325.0ms

Algorithm
intervals

setup69.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize16.0ms

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)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
3 → 94
Calls

3 calls. Slowest were:

19.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
7.0ms
(- (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))

series170.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

86.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
59.0ms
(- (exp x) (exp (- x)))
25.0ms
(+ (exp x) (exp (- x)))

simplify2.4s

Counts
91 → 103
Calls

91 calls. Slowest were:

298.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
251.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
116.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))

prune683.0ms

Pruning

3 alts after pruning (3 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 → 12
Calls

1 calls. Slowest were:

0.0ms
(tanh x)

series74.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

74.0ms
(tanh x)

simplify130.0ms

Counts
3 → 15
Calls

3 calls. Slowest were:

79.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
26.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
24.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune83.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize19.0ms

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)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 49
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))

series58.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

23.0ms
(- (+ 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))

simplify789.0ms

Counts
25 → 58
Calls

25 calls. Slowest were:

128.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
120.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
102.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))

prune735.0ms

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 → 189
Calls

4 calls. Slowest were:

11.0ms
(* (* (tanh x) (tanh x)) (tanh x))
7.0ms
(cbrt (* (* (tanh x) (tanh x)) (tanh x)))
0.0ms
(tanh x)

series386.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

114.0ms
(tanh x)
106.0ms
(tanh x)
89.0ms
(* (* (tanh x) (tanh x)) (tanh x))
77.0ms
(cbrt (* (* (tanh x) (tanh x)) (tanh x)))

simplify54.8s

Counts
204 → 201
Calls

204 calls. Slowest were:

903.0ms
(* (+ (exp x) (exp (- x))) (+ (exp (* 2 x)) 1))
831.0ms
(* (* (- (exp x) (exp (- x))) (- (exp (* 2 x)) 1)) (- (exp (* 2 x)) 1))
804.0ms
(* (* (+ (exp (* 2 x)) 1) (+ (exp (* 2 x)) 1)) (+ (exp (* 2 x)) 1))

prune2.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes25.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms