Average Error: 40.2 → 0.0
Time: 4.5s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{1}{\frac{x}{(e^{x} - 1)^*}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.2
Target39.3
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 40.2

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

  2. Simplified0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied associate-/l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))

Details

Time bar (total: 4.2s)Debug log

sample101.0ms

Algorithm
intervals

simplify12.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

12.0ms
(/ (- (exp x) 1) x)

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (expm1 x) x)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

3.0ms
(/ (expm1 x) x)

series24.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

24.0ms
(/ (expm1 x) x)

simplify193.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

121.0ms
(* -1 (/ (- 1 (exp x)) x))
61.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
8.0ms
(/ (- (exp x) 1) x)

prune157.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize9.0ms

Local error

Found 2 expressions with local error:

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

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

1.0ms
(/ 1 (/ x (expm1 x)))
0.0ms
(/ x (expm1 x))

series62.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(/ x (expm1 x))
27.0ms
(/ 1 (/ x (expm1 x)))

simplify217.0ms

Counts
15 → 41
Calls

15 calls. Slowest were:

106.0ms
(* -1 (/ (- 1 (exp x)) x))
50.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
39.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))

prune233.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize1.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (expm1 x) x)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

5.0ms
(/ (expm1 x) x)

series37.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

37.0ms
(/ (expm1 x) x)

simplify182.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

87.0ms
(* -1 (/ (- 1 (exp x)) x))
72.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
15.0ms
(/ (- (exp x) 1) x)

prune121.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes15.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.8s

Algorithm
intervals