Average Error: 39.8 → 0.4
Time: 9.3s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[e^{x} \cdot \frac{1}{(e^{x} - 1)^*}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.4
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.8

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{e^{x}}{(e^{x} - 1)^*}}\]
  3. Using strategy rm

  4. Applied div-inv0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019004 +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: 8.7s)Debug log

sample110.0ms

Algorithm
intervals

simplify5.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

5.0ms
(/ (exp x) (- (exp x) 1))

prune34.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize17.0ms

Local error

Found 1 expressions with local error:

0.5b
(/ (exp x) (expm1 x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

7.0ms
(/ (exp x) (expm1 x))

series39.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

39.0ms
(/ (exp x) (expm1 x))

simplify250.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

146.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
31.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
13.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune225.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.4b

localize7.0ms

Local error

Found 2 expressions with local error:

0.5b
(* (exp x) (/ 1 (expm1 x)))
0.0b
(/ 1 (expm1 x))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

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

series57.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

29.0ms
(/ 1 (expm1 x))
28.0ms
(* (exp x) (/ 1 (expm1 x)))

simplify635.0ms

Counts
22 → 47
Calls

22 calls. Slowest were:

494.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (/ 1 (expm1 x)) (/ 1 (expm1 x))) (/ 1 (expm1 x))))
38.0ms
(- (+ (* 1/12 x) (/ 1 x)) 1/2)
25.0ms
(* (exp x) (/ 1 (expm1 x)))

prune471.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.4b

localize7.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
0.5b
(/ (sqrt (exp x)) (expm1 x))
0.0b
(sqrt (exp x))
0.0b
(sqrt (exp x))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 113
Calls

4 calls. Slowest were:

7.0ms
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
4.0ms
(/ (sqrt (exp x)) (expm1 x))
2.0ms
(sqrt (exp x))

series130.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

64.0ms
(/ (sqrt (exp x)) (expm1 x))
39.0ms
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
14.0ms
(sqrt (exp x))
11.0ms
(sqrt (exp x))

simplify1.4s

Counts
99 → 125
Calls

99 calls. Slowest were:

381.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (/ (sqrt (exp x)) (expm1 x)) (/ (sqrt (exp x)) (expm1 x))) (/ (sqrt (exp x)) (expm1 x))))
243.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (expm1 x) (expm1 x)) (expm1 x))))
100.0ms
(- (+ (* 7/5760 (pow x 3)) (/ 1 x)) (* 1/24 x))

prune1.2s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.4b

localize13.0ms

Local error

Found 2 expressions with local error:

0.2b
(fma 1/12 x 1/2)
0.0b
(+ (fma 1/12 x 1/2) (/ 1 x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

7.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))
0.0ms
(fma 1/12 x 1/2)

series35.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

24.0ms
(fma 1/12 x 1/2)
11.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))

simplify98.0ms

Counts
12 → 35
Calls

12 calls. Slowest were:

20.0ms
(* (exp (fma 1/12 x 1/2)) (exp (/ 1 x)))
16.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))
10.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune224.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.4b

regimes46.0ms

Accuracy

0% (0.0b remaining)

Error of 0.4b against oracle of 0.3b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample3.8s

Algorithm
intervals