Average Error: 40.4 → 0.4
Time: 6.7s
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

Original40.4
Target40.0
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.4

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

  2. Simplified0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019002 +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: 6.4s)Debug log

sample154.0ms

Algorithm
intervals

simplify5.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 1 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

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

series56.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

55.0ms
(/ (exp x) (expm1 x))

simplify275.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

177.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
30.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
10.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune267.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 2 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
2 → 85
Calls

2 calls. Slowest were:

14.0ms
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
8.0ms
(/ (sqrt (exp x)) (expm1 x))

series109.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

83.0ms
(/ (sqrt (exp x)) (expm1 x))
26.0ms
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))

simplify1.2s

Counts
85 → 91
Calls

85 calls. Slowest were:

384.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (/ (sqrt (exp x)) (expm1 x)) (/ (sqrt (exp x)) (expm1 x))) (/ (sqrt (exp x)) (expm1 x))))
269.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (expm1 x) (expm1 x)) (expm1 x))))
148.0ms
(/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (expm1 x) (expm1 x)) (expm1 x)))

prune738.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes35.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample3.5s

Algorithm
intervals