Average Error: 39.5 → 0.0
Time: 5.9s
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

Original39.5
Target38.6
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 39.5

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

  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 2019008 +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: 5.6s)Debug log

sample74.0ms

Algorithm
intervals

simplify17.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

17.0ms
(/ (- (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)

series31.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

31.0ms
(/ (expm1 x) x)

simplify190.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

102.0ms
(* -1 (/ (- 1 (exp x)) x))
73.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
10.0ms
(/ (- (exp x) 1) x)

prune101.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize5.0ms

Local error

Found 2 expressions with local error:

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

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

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

series65.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(/ x (expm1 x))
30.0ms
(/ 1 (/ x (expm1 x)))

simplify217.0ms

Counts
15 → 41
Calls

15 calls. Slowest were:

111.0ms
(* -1 (/ (- 1 (exp x)) x))
47.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
38.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))

prune272.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 3 expressions with local error:

0.7b
(expm1 (/ (expm1 x) x))
0.0b
(/ (expm1 x) x)
0.0b
(log1p (expm1 (/ (expm1 x) x)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

6.0ms
(/ (expm1 x) x)
1.0ms
(expm1 (/ (expm1 x) x))
0.0ms
(log1p (expm1 (/ (expm1 x) x)))

series317.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

250.0ms
(log1p (expm1 (/ (expm1 x) x)))
35.0ms
(expm1 (/ (expm1 x) x))
31.0ms
(/ (expm1 x) x)

simplify1.2s

Counts
14 → 47
Calls

14 calls. Slowest were:

362.0ms
(- (+ (* 7/24 (* (pow x 2) E)) (+ E (* 1/2 (* x E)))) 1)
287.0ms
(- (+ (/ (exp x) x) (+ (* 1/2 (/ 1 (pow x 2))) (+ (* 1/6 (/ (pow (exp x) 3) (pow x 3))) (+ (* 1/2 (/ (pow (exp x) 2) (pow x 2))) (* 1/2 (/ (exp x) (pow x 3))))))) (+ (/ (exp x) (pow x 2)) (+ (* 1/2 (/ (pow (exp x) 2) (pow x 3))) (+ (* 1/6 (/ 1 (pow x 3))) (/ 1 x)))))
204.0ms
(- (+ (/ (exp x) x) (+ (* 1/2 (/ 1 (pow x 2))) (+ (* 1/6 (/ (pow (exp x) 3) (pow x 3))) (+ (* 1/2 (/ (pow (exp x) 2) (pow x 2))) (* 1/2 (/ (exp x) (pow x 3))))))) (+ (/ (exp x) (pow x 2)) (+ (* 1/2 (/ (pow (exp x) 2) (pow x 3))) (+ (* 1/6 (/ 1 (pow x 3))) (/ 1 x)))))

prune346.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes31.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.7s

Algorithm
intervals