Average Error: 57.9 → 0.0
Time: 17.0s
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 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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 16.7s)Debug log

sample359.0ms

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)

simplify22.0ms

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

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.7b

localize27.0ms

Local error

Found 2 expressions with local error:

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

rewrite39.0ms

Algorithm
rewrite-expression-head
Rules
44×*-un-lft-identity
32×add-sqr-sqrt
20×add-cube-cbrt
20×times-frac
9×prod-diff
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*
3×fma-neg
2×associate-/l/
2×flip--
2×log1p-expm1-u
2×associate-/r/
2×flip3--
2×pow1
2×expm1-log1p-u
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 → 77
Calls
2 calls:
Slowest
28.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
9.0ms
(- (exp x) (exp (- x)))

series155.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
99.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
56.0ms
(- (exp x) (exp (- x)))

simplify2.0s

Counts
84 → 83
Calls
84 calls:
Slowest
318.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
226.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
120.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))
91.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
81.0ms
(fma 1 (exp x) (- (* (exp (- x)) 1)))

prune616.0ms

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)

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)

series89.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
89.0ms
(tanh x)

simplify153.0ms

Counts
3 → 15
Calls
3 calls:
Slowest
105.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
25.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
22.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune94.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize33.0ms

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
Rules
8×associate-*r*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×add-log-exp
3×log1p-expm1-u
3×add-exp-log
3×prod-diff
3×fma-neg
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×*-commutative
1×flip--
1×associate--l+
1×unpow3
1×diff-log
1×flip3--
1×sub-neg
1×cube-mult
Counts
3 → 49
Calls
3 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))

series74.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
33.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
24.0ms
(* 2/15 (pow x 5))
17.0ms
(* 1/3 (pow x 3))

simplify1.0s

Counts
25 → 58
Calls
25 calls:
Slowest
146.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))
133.0ms
(fma (sqrt (+ x (* 2/15 (pow x 5)))) (sqrt (+ x (* 2/15 (pow x 5)))) (- (* (pow x 3) 1/3)))
131.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
126.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
126.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))

prune559.0ms

Pruning

2 alts after pruning (0 fresh and 2 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

end0.0ms

sample11.4s

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)