Average Error: 39.9 → 0.5
Time: 9.6s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{(e^{x} - 1)^*}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.4
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.9

    \[\frac{e^{x}}{e^{x} - 1}\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{e^{x}}{(e^{x} - 1)^*}}\]

  3. Taylor expanded around -inf 39.9

    \[\leadsto \color{blue}{\frac{e^{x}}{e^{x} - 1}}\]

  4. Simplified0.5

    \[\leadsto \color{blue}{\frac{e^{x}}{(e^{x} - 1)^*}}\]

  5. Final simplification0.5

    \[\leadsto \frac{e^{x}}{(e^{x} - 1)^*}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))

Details

Time bar (total: 9.0s)Debug log

sample142.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

3.0ms
(/ (exp x) (- (exp x) 1))

prune9.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize8.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ (exp x) (expm1 x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

3.0ms
(/ (exp x) (expm1 x))

series35.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

35.0ms
(/ (exp x) (expm1 x))

simplify250.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

151.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
22.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
10.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune300.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.2b

localize1.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ (exp x) (expm1 x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

6.0ms
(/ (exp x) (expm1 x))

series36.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

36.0ms
(/ (exp x) (expm1 x))

simplify245.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

142.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
20.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
18.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))

prune237.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.2b

localize5.0ms

Local error

Found 2 expressions with local error:

0.2b
(* (exp x) (/ 1 (expm1 x)))
0.0b
(/ 1 (expm1 x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

4.0ms
(* (exp x) (/ 1 (expm1 x)))
1.0ms
(/ 1 (expm1 x))

series63.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

37.0ms
(* (exp x) (/ 1 (expm1 x)))
26.0ms
(/ 1 (expm1 x))

simplify487.0ms

Counts
22 → 47
Calls

22 calls. Slowest were:

391.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (/ 1 (expm1 x)) (/ 1 (expm1 x))) (/ 1 (expm1 x))))
28.0ms
(- (+ (* 1/12 x) (/ 1 x)) 1/2)
13.0ms
(* (exp x) (/ 1 (expm1 x)))

prune333.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.2b

localize10.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (cbrt (exp x)) (expm1 x))
0.3b
(* (* (cbrt (exp x)) (cbrt (exp x))) (/ (cbrt (exp x)) (expm1 x)))
0.2b
(cbrt (exp x))
0.2b
(cbrt (exp x))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 111
Calls

4 calls. Slowest were:

16.0ms
(* (* (cbrt (exp x)) (cbrt (exp x))) (/ (cbrt (exp x)) (expm1 x)))
5.0ms
(/ (cbrt (exp x)) (expm1 x))
3.0ms
(cbrt (exp x))

series470.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

171.0ms
(/ (cbrt (exp x)) (expm1 x))
133.0ms
(cbrt (exp x))
130.0ms
(cbrt (exp x))
35.0ms
(* (* (cbrt (exp x)) (cbrt (exp x))) (/ (cbrt (exp x)) (expm1 x)))

simplify1.4s

Counts
98 → 123
Calls

98 calls. Slowest were:

248.0ms
(* (* (* (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (exp x)) (cbrt (exp x)))) (* (cbrt (exp x)) (cbrt (exp x)))) (* (* (/ (cbrt (exp x)) (expm1 x)) (/ (cbrt (exp x)) (expm1 x))) (/ (cbrt (exp x)) (expm1 x))))
229.0ms
(* (* (* (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (exp x)) (cbrt (exp x)))) (* (cbrt (exp x)) (cbrt (exp x)))) (/ (exp x) (* (* (expm1 x) (expm1 x)) (expm1 x))))
124.0ms
(* (* (exp x) (exp x)) (/ (exp x) (* (* (expm1 x) (expm1 x)) (expm1 x))))

prune1.1s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.2b

regimes25.0ms

Accuracy

0% (0.1b remaining)

Error of 0.5b against oracle of 0.4b and baseline of 0.5b

bsearch0.0ms

end0.0ms

sample3.8s

Algorithm
intervals