Average Error: 58.2 → 0.0
Time: 21.1s
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 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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 20.7s)Debug log

sample304.0ms

Algorithm
intervals
Results
153.0ms229×body10240exit
84.0ms111×body1280valid
37.0ms78×body640valid
11.0ms36×body320valid
3.0ms15×body160valid
2.0ms16×body80valid

simplify62.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
62.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.9b

localize15.0ms

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)))

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
48×*-un-lft-identity
34×add-sqr-sqrt
22×add-cube-cbrt
20×times-frac
9×prod-diff
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×log1p-expm1-u
3×fma-neg
3×fma-def
3×pow1
3×expm1-log1p-u
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 → 94
Calls
3 calls:
Slowest
30.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
5.0ms
(- (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))

series161.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
84.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
51.0ms
(- (exp x) (exp (- x)))
26.0ms
(+ (exp x) (exp (- x)))

simplify2.2s

Counts
91 → 103
Calls
91 calls:
Slowest
330.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
191.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
110.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
104.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))
86.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

prune753.0ms

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
Rules
1×add-log-exp
1×tanh-def
1×log1p-expm1-u
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 12
Calls
1 calls:
Slowest
0.0ms
(tanh x)

series114.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
114.0ms
(tanh x)

simplify172.0ms

Counts
3 → 15
Calls
3 calls:
Slowest
87.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
47.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
36.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune119.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize17.0ms

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)))

rewrite16.0ms

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×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
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 → 93
Calls
4 calls:
Slowest
11.0ms
(* (cbrt (tanh x)) (cbrt (tanh x)))
1.0ms
(cbrt (tanh x))
1.0ms
(cbrt (tanh x))
1.0ms
(cbrt (tanh x))

series995.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
255.0ms
(cbrt (tanh x))
250.0ms
(cbrt (tanh x))
247.0ms
(cbrt (tanh x))
243.0ms
(* (cbrt (tanh x)) (cbrt (tanh x)))

simplify2.2s

Counts
74 → 105
Calls
74 calls:
Slowest
147.0ms
(pow (/ (pow (- (exp x) (exp (- x))) 2) (pow (+ (exp x) (exp (- x))) 2)) 1/3)
130.0ms
(* (cbrt (- (exp (* 2 x)) 1)) (cbrt (tanh x)))
125.0ms
(pow (/ (pow (- (exp x) (exp (* -1 x))) 2) (pow (+ (exp (* -1 x)) (exp x)) 2)) 1/3)
103.0ms
(* (cbrt (- 1 (exp (* -2 x)))) (cbrt (tanh x)))
75.0ms
(* (cbrt (- (exp x) (exp (- x)))) (cbrt (tanh x)))

prune1.0s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize27.0ms

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)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×associate-*r*
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×prod-diff
3×fma-neg
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 → 61
Calls
4 calls:
Slowest
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))
1.0ms
(+ x (* 2/15 (pow x 5)))

series142.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(+ x (* 2/15 (pow x 5)))
30.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
19.0ms
(* 2/15 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify985.0ms

Counts
28 → 73
Calls
28 calls:
Slowest
177.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
142.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
128.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))
116.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
93.0ms
(fma (sqrt (+ x (* 2/15 (pow x 5)))) (sqrt (+ x (* 2/15 (pow x 5)))) (- (* (pow x 3) 1/3)))

prune615.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes23.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample10.7s

Algorithm
intervals
Results
5.9s7501×body10240exit
2.9s3314×body1280valid
1.2s2413×body640valid
402.0ms1277×body320valid
106.0ms580×body160valid
38.0ms416×body80valid