Average Error: 57.9 → 0.0
Time: 15.1s
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 2019016 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 14.9s)Debug log

sample296.0ms

Algorithm
intervals
Results
521×(pre true 80)
264×(body exit 10240)
105×(body real 1280)
80×(body real 640)
42×(body real 320)
15×(body real 80)
15×(body real 160)

simplify34.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
34.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.7b

localize58.0ms

Local error

Found 2 expressions with local error:

5.9b
(- (exp x) (exp (- x)))
0.5b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
37×*-un-lft-identity
25×add-sqr-sqrt
20×times-frac
13×add-cube-cbrt
6×difference-of-squares
6×distribute-lft-out--
6×distribute-lft-out
5×associate-/l*
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×associate-/r*
2×associate-/l/
2×flip--
2×associate-/r/
2×flip3--
2×pow1
1×div-inv
1×flip-+
1×tanh-undef
1×div-exp
1×div-sub
1×diff-log
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
2 → 61
Calls
2 calls:
Slowest
30.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
8.0ms
(- (exp x) (exp (- x)))

series158.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
116.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
42.0ms
(- (exp x) (exp (- x)))

simplify1.9s

Counts
64 → 67
Calls
64 calls:
Slowest
442.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
303.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
211.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
159.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))
57.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune489.0ms

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
Rules
1×add-log-exp
1×tanh-def
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(tanh x)

series103.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
103.0ms
(tanh x)

simplify113.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
66.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
25.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
21.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune100.0ms

Pruning

2 alts after pruning (1 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)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×add-exp-log
3×add-cbrt-cube
3×pow1
2×*-commutative
1×flip--
1×associate--l+
1×unpow3
1×diff-log
1×flip3--
1×sub-neg
1×cube-mult
Counts
3 → 37
Calls
3 calls:
Slowest
8.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
2.0ms
(* 1/3 (pow x 3))
2.0ms
(* 2/15 (pow x 5))

series89.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
42.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
26.0ms
(* 1/3 (pow x 3))
20.0ms
(* 2/15 (pow x 5))

simplify360.0ms

Counts
17 → 46
Calls
17 calls:
Slowest
133.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
69.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
68.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
58.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
5.0ms
(* 1/3 (pow x 3))

prune444.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes18.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample10.6s

Algorithm
intervals
Results
15571×(pre true 80)
7571×(body exit 10240)
3271×(body real 1280)
2451×(body real 640)
1225×(body real 320)
616×(body real 160)
437×(body real 80)