Average Error: 58.3 → 0.0
Time: 14.1s
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 58.3

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019010 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 13.8s)Debug log

sample283.0ms

Algorithm
intervals

simplify23.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
23.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize17.0ms

Local error

Found 2 expressions with local error:

5.4b
(- (exp x) (exp (- x)))
0.7b
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

rewrite32.0ms

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
26.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
5.0ms
(- (exp x) (exp (- x)))

series134.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
88.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
46.0ms
(- (exp x) (exp (- x)))

simplify1.9s

Counts
64 → 67
Calls
64 calls:
Slowest
437.0ms
(/ (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
325.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
186.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))
145.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
54.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune416.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize5.0ms

Local error

Found 1 expressions with local error:

0.0b
(tanh x)

rewrite1.0ms

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)

series116.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
116.0ms
(tanh x)

simplify107.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
49.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
29.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
28.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))

prune70.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

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

rewrite8.0ms

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
4.0ms
(- (+ 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)))

series126.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
57.0ms
(+ x (* 2/15 (pow x 5)))
31.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
22.0ms
(* 2/15 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify330.0ms

Counts
20 → 59
Calls
20 calls:
Slowest
121.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
61.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
59.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
49.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
6.0ms
(* 1/3 (pow x 3))

prune528.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes32.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.7s

Algorithm
intervals