Average Error: 39.9 → 0.4
Time: 20.2s
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

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

Derivation

  1. Initial program 39.9

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

  2. Initial simplification0.4

    \[\leadsto \frac{e^{x}}{(e^{x} - 1)^*}\]
  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 2018362 +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: 12.7s)Debug log

start137.0ms

Algorithm
intervals

setup10.0ms

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))

series43.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

43.0ms
(/ (exp x) (expm1 x))

simplify189.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

86.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
19.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
13.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))

prune295.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.7b

localize8.0ms

Local error

Found 2 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

8.0ms
(* (exp x) (/ 1 (expm1 x)))
2.0ms
(/ 1 (expm1 x))

series47.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

29.0ms
(* (exp x) (/ 1 (expm1 x)))
17.0ms
(/ 1 (expm1 x))

simplify669.0ms

Counts
22 → 47
Calls

22 calls. Slowest were:

573.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (/ 1 (expm1 x)) (/ 1 (expm1 x))) (/ 1 (expm1 x))))
24.0ms
(- (+ (* 1/12 x) (/ 1 x)) 1/2)
14.0ms
(* (exp x) (/ 1 (expm1 x)))

prune351.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

localize9.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (sqrt (exp x)) (/ (expm1 x) (sqrt (exp x))))
0.6b
(/ (expm1 x) (sqrt (exp x)))
0.2b
(sqrt (exp x))
0.2b
(sqrt (exp x))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 247
Calls

4 calls. Slowest were:

6.0ms
(/ (sqrt (exp x)) (/ (expm1 x) (sqrt (exp x))))
4.0ms
(/ (expm1 x) (sqrt (exp x)))
1.0ms
(sqrt (exp x))

series110.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(/ (expm1 x) (sqrt (exp x)))
21.0ms
(/ (sqrt (exp x)) (/ (expm1 x) (sqrt (exp x))))
11.0ms
(sqrt (exp x))
11.0ms
(sqrt (exp x))

simplify7.8s

Counts
363 → 259
Calls

363 calls. Slowest were:

511.0ms
(/ (sqrt 1) (/ (* (cbrt (expm1 x)) (cbrt (expm1 x))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))))
469.0ms
(/ (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x)))) (/ 1 (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))))
438.0ms
(/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (/ (expm1 x) (sqrt (exp x))) (/ (expm1 x) (sqrt (exp x)))) (/ (expm1 x) (sqrt (exp x)))))

prune2.5s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.7b

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)

series37.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

24.0ms
(fma 1/12 x 1/2)
13.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))

simplify112.0ms

Counts
12 → 35
Calls

12 calls. Slowest were:

25.0ms
(* (exp (fma 1/12 x 1/2)) (exp (/ 1 x)))
20.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
11.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune295.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.7b

regimes58.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms