Average Error: 39.6 → 0.0
Time: 6.9s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[(e^{\log_* (1 + \frac{(e^{x} - 1)^*}{x})} - 1)^*\]

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. Using strategy rm

  4. Applied expm1-log1p-u0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 +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: 6.6s)Debug log

sample64.0ms

Algorithm
intervals

simplify13.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

13.0ms
(/ (- (exp x) 1) x)

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (expm1 x) x)

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

5.0ms
(/ (expm1 x) x)

series27.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

27.0ms
(/ (expm1 x) x)

simplify163.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

109.0ms
(* -1 (/ (- 1 (exp x)) x))
44.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
7.0ms
(/ (- (exp x) 1) x)

prune172.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 3 expressions with local error:

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

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

3.0ms
(/ (expm1 x) x)
1.0ms
(log1p (/ (expm1 x) x))
0.0ms
(expm1 (log1p (/ (expm1 x) x)))

series313.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

237.0ms
(expm1 (log1p (/ (expm1 x) x)))
52.0ms
(log1p (/ (expm1 x) x))
24.0ms
(/ (expm1 x) x)

simplify726.0ms

Counts
14 → 47
Calls

14 calls. Slowest were:

201.0ms
(- (+ (/ (exp x) x) (+ (/ (exp x) (pow x 2)) (+ (* 1/3 (/ (pow (exp x) 3) (pow x 3))) (/ (exp x) (pow x 3))))) (+ (/ (pow (exp x) 2) (pow x 3)) (+ (* 1/2 (/ 1 (pow x 2))) (+ (* 1/2 (/ (pow (exp x) 2) (pow x 2))) (+ (* 1/3 (/ 1 (pow x 3))) (/ 1 x))))))
162.0ms
(- (+ (/ (exp x) x) (+ (/ (exp x) (pow x 2)) (+ (* 1/3 (/ (pow (exp x) 3) (pow x 3))) (/ (exp x) (pow x 3))))) (+ (/ (pow (exp x) 2) (pow x 3)) (+ (* 1/2 (/ 1 (pow x 2))) (+ (* 1/2 (/ (pow (exp x) 2) (pow x 2))) (+ (* 1/3 (/ 1 (pow x 3))) (/ 1 x))))))
86.0ms
(* -1 (/ (- 1 (exp x)) x))

prune382.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (sqrt (expm1 (log1p (/ (expm1 x) x)))) (sqrt (expm1 (log1p (/ (expm1 x) x)))))
0.1b
(sqrt (expm1 (log1p (/ (expm1 x) x))))
0.1b
(sqrt (expm1 (log1p (/ (expm1 x) x))))
0.0b
(/ (expm1 x) x)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (expm1 (log1p (/ (expm1 x) x)))) (sqrt (expm1 (log1p (/ (expm1 x) x)))))
3.0ms
(/ (expm1 x) x)
1.0ms
(sqrt (expm1 (log1p (/ (expm1 x) x))))

series456.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

242.0ms
(* (sqrt (expm1 (log1p (/ (expm1 x) x)))) (sqrt (expm1 (log1p (/ (expm1 x) x)))))
90.0ms
(sqrt (expm1 (log1p (/ (expm1 x) x))))
87.0ms
(sqrt (expm1 (log1p (/ (expm1 x) x))))
36.0ms
(/ (expm1 x) x)

simplify772.0ms

Counts
45 → 88
Calls

45 calls. Slowest were:

254.0ms
(* (* (* (sqrt (expm1 (log1p (/ (expm1 x) x)))) (sqrt (expm1 (log1p (/ (expm1 x) x))))) (sqrt (expm1 (log1p (/ (expm1 x) x))))) (* (* (sqrt (expm1 (log1p (/ (expm1 x) x)))) (sqrt (expm1 (log1p (/ (expm1 x) x))))) (sqrt (expm1 (log1p (/ (expm1 x) x))))))
137.0ms
(* -1 (/ (- 1 (exp x)) x))
102.0ms
(* -1 (/ (- 1 (exp x)) x))

prune863.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes36.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.5s

Algorithm
intervals