Average Error: 58.2 → 0.0
Time: 16.8s
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 58.2

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019022 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 16.5s)Debug log

sample306.0ms

Algorithm
intervals
Results
162.0ms229×body10240exit
78.0ms111×body1280valid
38.0ms78×body640valid
12.0ms36×body320valid
3.0ms15×body160valid
2.0ms16×body80valid

simplify22.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
22.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.9b

localize19.0ms

Local error

Found 3 expressions with local error:

6.4b
(- (exp x) (exp (- x)))
1.5b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
40×*-un-lft-identity
26×add-sqr-sqrt
20×times-frac
14×add-cube-cbrt
7×add-log-exp
7×distribute-lft-out
6×difference-of-squares
6×distribute-lft-out--
5×associate-/l*
5×add-exp-log
5×add-cbrt-cube
4×associate-/r*
3×pow1
2×associate-/l/
2×flip--
2×flip-+
2×associate-/r/
2×flip3--
2×flip3-+
1×div-inv
1×tanh-undef
1×div-exp
1×div-sub
1×diff-log
1×frac-2neg
1×sub-neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
3 → 73
Calls
3 calls:
Slowest
27.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
4.0ms
(+ (exp x) (exp (- x)))
4.0ms
(- (exp x) (exp (- x)))

series177.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
104.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
44.0ms
(- (exp x) (exp (- x)))
29.0ms
(+ (exp x) (exp (- x)))

simplify1.7s

Counts
69 → 82
Calls
69 calls:
Slowest
414.0ms
(/ (* (* (- (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))))
123.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))
109.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
53.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune524.0ms

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
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)

series86.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
86.0ms
(tanh x)

simplify111.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
50.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
41.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
20.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune65.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize10.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (tanh x))
0.6b
(cbrt (tanh x))
0.6b
(cbrt (tanh x))
0.5b
(* (cbrt (tanh x)) (cbrt (tanh x)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
33×cbrt-div
15×cbrt-prod
11×tanh-def
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
9×frac-times
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×add-cbrt-cube
3×associate-*r/
3×associate-*l/
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 85
Calls
4 calls:
Slowest
5.0ms
(* (cbrt (tanh x)) (cbrt (tanh x)))
1.0ms
(cbrt (tanh x))
0.0ms
(cbrt (tanh x))
0.0ms
(cbrt (tanh x))

series914.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
260.0ms
(* (cbrt (tanh x)) (cbrt (tanh x)))
233.0ms
(cbrt (tanh x))
216.0ms
(cbrt (tanh x))
205.0ms
(cbrt (tanh x))

simplify1.8s

Counts
74 → 97
Calls
74 calls:
Slowest
143.0ms
(pow (/ (pow (- (exp x) (exp (* -1 x))) 2) (pow (+ (exp (* -1 x)) (exp x)) 2)) 1/3)
97.0ms
(pow (/ (pow (- (exp x) (exp (- x))) 2) (pow (+ (exp x) (exp (- x))) 2)) 1/3)
79.0ms
(* (cbrt (- (exp (* 2 x)) 1)) (cbrt (tanh x)))
66.0ms
(* (cbrt (- (exp x) (exp (- x)))) (cbrt (tanh x)))
65.0ms
(* (cbrt (- 1 (exp (* -2 x)))) (cbrt (tanh x)))

prune645.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 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)))
0.0b
(+ x (* 2/15 (pow x 5)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
6×add-log-exp
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-exp-log
4×add-cbrt-cube
4×pow1
2×*-commutative
1×flip--
1×flip-+
1×associate--l+
1×unpow3
1×diff-log
1×flip3--
1×flip3-+
1×sub-neg
1×+-commutative
1×cube-mult
Counts
4 → 47
Calls
4 calls:
Slowest
7.0ms
(- (+ 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))
1.0ms
(+ x (* 2/15 (pow x 5)))

series124.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
66.0ms
(+ x (* 2/15 (pow x 5)))
24.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
18.0ms
(* 2/15 (pow x 5))
15.0ms
(* 1/3 (pow x 3))

simplify329.0ms

Counts
20 → 59
Calls
20 calls:
Slowest
102.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
75.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
59.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
55.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
5.0ms
(* 1/3 (pow x 3))

prune421.0ms

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

end0.0ms

sample9.2s

Algorithm
intervals
Results
5.2s7501×body10240exit
2.3s3314×body1280valid
947.0ms2413×body640valid
352.0ms1277×body320valid
94.0ms580×body160valid
36.0ms416×body80valid