Average Error: 58.3 → 0.0
Time: 16.4s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 16.2s)Debug log

sample333.0ms

Algorithm
intervals

simplify33.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
33.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize15.0ms

Local error

Found 2 expressions with local error:

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

rewrite31.0ms

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
25.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
5.0ms
(- (exp x) (exp (- x)))

series141.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
86.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))
55.0ms
(- (exp x) (exp (- x)))

simplify2.2s

Counts
84 → 83
Calls
84 calls:
Slowest
379.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
198.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
115.0ms
(- (pow (exp x) 3) (pow (exp (- x)) 3))
91.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
76.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune496.0ms

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)

series106.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
106.0ms
(tanh x)

simplify206.0ms

Counts
3 → 15
Calls
3 calls:
Slowest
128.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
49.0ms
(/ (- (exp x) (exp (* -1 x))) (+ (exp (* -1 x)) (exp x)))
28.0ms
(/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x))))

prune79.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

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)))

rewrite18.0ms

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
11.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
3.0ms
(* 1/3 (pow x 3))
2.0ms
(* 2/15 (pow x 5))
2.0ms
(+ x (* 2/15 (pow x 5)))

series132.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
50.0ms
(+ x (* 2/15 (pow x 5)))
41.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
25.0ms
(* 2/15 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify930.0ms

Counts
28 → 73
Calls
28 calls:
Slowest
151.0ms
(fma 1 (+ x (* 2/15 (pow x 5))) (- (* (pow x 3) 1/3)))
119.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
109.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))
105.0ms
(/ (exp (+ x (* 2/15 (pow x 5)))) (exp (* 1/3 (pow x 3))))
100.0ms
(- (+ x (* 2/15 (pow x 5))) (* 1/3 (pow x 3)))

prune683.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes28.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample10.7s

Algorithm
intervals