Average Error: 39.6 → 0.0
Time: 9.2s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{(e^{x} - 1)^*}{x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target38.7
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 39.6

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019005 +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: 8.8s)Debug log

sample66.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

7.0ms
(/ (- (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)

series22.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

22.0ms
(/ (expm1 x) x)

simplify161.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

92.0ms
(* -1 (/ (- 1 (exp x)) x))
50.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
9.0ms
(/ (exp x) x)

prune168.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize10.0ms

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))

series58.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

32.0ms
(/ x (expm1 x))
26.0ms
(/ 1 (/ x (expm1 x)))

simplify296.0ms

Counts
15 → 41
Calls

15 calls. Slowest were:

132.0ms
(* -1 (/ (- 1 (exp x)) x))
75.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
58.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))

prune229.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x))))
0.0b
(/ 1 (* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x)))))
0.0b
(/ x (expm1 x))
0.0b
(/ x (expm1 x))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

7.0ms
(* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x))))
4.0ms
(/ 1 (* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x)))))
0.0ms
(/ x (expm1 x))

series162.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

46.0ms
(/ x (expm1 x))
43.0ms
(/ x (expm1 x))
41.0ms
(/ 1 (* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x)))))
32.0ms
(* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x))))

simplify623.0ms

Counts
57 → 105
Calls

57 calls. Slowest were:

188.0ms
(* (* (* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x)))) (sqrt (/ x (expm1 x)))) (* (* (sqrt (/ x (expm1 x))) (sqrt (/ x (expm1 x)))) (sqrt (/ x (expm1 x)))))
92.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
86.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))

prune827.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.2b
(cbrt (/ (expm1 x) x))
0.2b
(cbrt (/ (expm1 x) x))
0.2b
(cbrt (/ (expm1 x) x))
0.2b
(* (cbrt (/ (expm1 x) x)) (cbrt (/ (expm1 x) x)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (/ (expm1 x) x)) (cbrt (/ (expm1 x) x)))
1.0ms
(cbrt (/ (expm1 x) x))
1.0ms
(cbrt (/ (expm1 x) x))

series763.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

207.0ms
(cbrt (/ (expm1 x) x))
201.0ms
(cbrt (/ (expm1 x) x))
179.0ms
(cbrt (/ (expm1 x) x))
176.0ms
(* (cbrt (/ (expm1 x) x)) (cbrt (/ (expm1 x) x)))

simplify1.8s

Counts
53 → 92
Calls

53 calls. Slowest were:

273.0ms
(exp (* 1/3 (+ (* 2 (log (/ 1 x))) (log (pow (- (exp x) 1) 2)))))
220.0ms
(* (pow (cbrt -1) 2) (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (log (pow (- (exp x) 1) 2))))))
213.0ms
(* (exp (* 1/3 (+ (log (- (exp x) 1)) (log (/ -1 x))))) (cbrt -1))

prune692.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes19.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample2.8s

Algorithm
intervals