Average Error: 39.7 → 0.4
Time: 7.4s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{1}{\frac{(e^{x} - 1)^*}{e^{x}}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.7
Target39.2
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.7

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

  2. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019008 +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: 7.1s)Debug log

sample98.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize7.0ms

Local error

Found 1 expressions with local error:

0.7b
(/ (exp x) (expm1 x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

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

series33.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

33.0ms
(/ (exp x) (expm1 x))

simplify224.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

126.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
24.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
18.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune304.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.7b

localize5.0ms

Local error

Found 2 expressions with local error:

0.7b
(/ (expm1 x) (exp x))
0.0b
(/ 1 (/ (expm1 x) (exp x)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 60
Calls

2 calls. Slowest were:

3.0ms
(/ (expm1 x) (exp x))
1.0ms
(/ 1 (/ (expm1 x) (exp x)))

series55.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

34.0ms
(/ 1 (/ (expm1 x) (exp x)))
21.0ms
(/ (expm1 x) (exp x))

simplify603.0ms

Counts
47 → 66
Calls

47 calls. Slowest were:

326.0ms
(- (+ x (* 1/6 (pow x 3))) (* 1/2 (pow x 2)))
105.0ms
(/ (* (* (expm1 x) (expm1 x)) (expm1 x)) (* (* (exp x) (exp x)) (exp x)))
31.0ms
(/ (* (cbrt (expm1 x)) (cbrt (expm1 x))) (* (cbrt (exp x)) (cbrt (exp x))))

prune569.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize19.0ms

Local error

Found 2 expressions with local error:

0.1b
(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)

series36.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

24.0ms
(fma 1/12 x 1/2)
12.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))

simplify151.0ms

Counts
12 → 35
Calls

12 calls. Slowest were:

41.0ms
(* (exp (fma 1/12 x 1/2)) (exp (/ 1 x)))
20.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
20.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune289.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 3 expressions with local error:

3.7b
(- 1 (/ 1 (exp x)))
0.0b
(/ 1 (exp x))
0.0b
(/ 1 (- 1 (/ 1 (exp x))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 51
Calls

3 calls. Slowest were:

3.0ms
(/ 1 (- 1 (/ 1 (exp x))))
2.0ms
(- 1 (/ 1 (exp x)))
1.0ms
(/ 1 (exp x))

series80.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

37.0ms
(/ 1 (- 1 (/ 1 (exp x))))
28.0ms
(- 1 (/ 1 (exp x)))
14.0ms
(/ 1 (exp x))

simplify573.0ms

Counts
19 → 60
Calls

19 calls. Slowest were:

321.0ms
(- (+ x (* 1/6 (pow x 3))) (* 1/2 (pow x 2)))
92.0ms
(+ (* 1 1) (+ (* (/ 1 (exp x)) (/ 1 (exp x))) (* 1 (/ 1 (exp x)))))
56.0ms
(- (+ (* 1/2 (pow x 2)) 1) x)

prune428.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes67.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch1.0ms

end0.0ms

sample3.5s

Algorithm
intervals