Average Error: 40.4 → 0.5
Time: 14.9s
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
Target39.9
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.4

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

  2. Initial simplification0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2018360 +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: 7.7s)Debug log

start152.0ms

Algorithm
intervals

setup16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize44.0ms

Local error

Found 1 expressions with local error:

1.0b
(/ (exp x) (expm1 x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

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

series25.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

25.0ms
(/ (exp x) (expm1 x))

simplify228.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

118.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
21.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
14.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))

prune271.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.9b

localize30.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ (cbrt (exp x)) (expm1 x))
1.0b
(* (* (cbrt (exp x)) (cbrt (exp x))) (/ (cbrt (exp x)) (expm1 x)))
0.0b
(* (cbrt (exp x)) (cbrt (exp x)))
0.0b
(cbrt (exp x))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 128
Calls

4 calls. Slowest were:

13.0ms
(* (* (cbrt (exp x)) (cbrt (exp x))) (/ (cbrt (exp x)) (expm1 x)))
5.0ms
(* (cbrt (exp x)) (cbrt (exp x)))
4.0ms
(/ (cbrt (exp x)) (expm1 x))

series420.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

147.0ms
(/ (cbrt (exp x)) (expm1 x))
129.0ms
(* (cbrt (exp x)) (cbrt (exp x)))
121.0ms
(cbrt (exp x))
23.0ms
(* (* (cbrt (exp x)) (cbrt (exp x))) (/ (cbrt (exp x)) (expm1 x)))

simplify1.7s

Counts
115 → 140
Calls

115 calls. Slowest were:

421.0ms
(* (* (exp x) (exp x)) (* (* (/ (cbrt (exp x)) (expm1 x)) (/ (cbrt (exp x)) (expm1 x))) (/ (cbrt (exp x)) (expm1 x))))
281.0ms
(* (* (* (* (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))))
158.0ms
(* (* (* (* (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))))

prune1.4s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.9b

localize16.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ (cbrt (exp x)) (expm1 x))
1.0b
(* (cbrt (* (exp x) (exp x))) (/ (cbrt (exp x)) (expm1 x)))
0.0b
(* (exp x) (exp x))
0.0b
(cbrt (* (exp x) (exp x)))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

16.0ms
(* (cbrt (* (exp x) (exp x))) (/ (cbrt (exp x)) (expm1 x)))
7.0ms
(/ (cbrt (exp x)) (expm1 x))
7.0ms
(* (exp x) (exp x))

series394.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

190.0ms
(/ (cbrt (exp x)) (expm1 x))
170.0ms
(cbrt (* (exp x) (exp x)))
23.0ms
(* (cbrt (* (exp x) (exp x))) (/ (cbrt (exp x)) (expm1 x)))
10.0ms
(* (exp x) (exp x))

simplify1.0s

Counts
101 → 128
Calls

101 calls. Slowest were:

488.0ms
(* (* (exp x) (exp x)) (* (* (/ (cbrt (exp x)) (expm1 x)) (/ (cbrt (exp x)) (expm1 x))) (/ (cbrt (exp x)) (expm1 x))))
91.0ms
(* (* (exp x) (exp x)) (/ (exp x) (* (* (expm1 x) (expm1 x)) (expm1 x))))
57.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))

prune1.4s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.9b

localize12.0ms

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))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

4.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))
0.0ms
(fma 1/12 x 1/2)

series38.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

25.0ms
(fma 1/12 x 1/2)
13.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))

simplify122.0ms

Counts
12 → 35
Calls

12 calls. Slowest were:

33.0ms
(* (exp (fma 1/12 x 1/2)) (exp (/ 1 x)))
12.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))
10.0ms
(* (cbrt (fma 1/12 x 1/2)) (cbrt (fma 1/12 x 1/2)))

prune338.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.9b

regimes31.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms