Average Error: 39.7 → 0.1
Time: 12.5s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\log_* (1 + (e^{\frac{(e^{x} - 1)^*}{x}} - 1)^*)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.7
Target38.9
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 39.7

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

  2. Initial simplification0.0

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

  4. Applied log1p-expm1-u0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2018362 +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: 7.6s)Debug log

start130.0ms

Algorithm
intervals

setup30.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

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:

6.0ms
(/ (expm1 x) x)

series33.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

33.0ms
(/ (expm1 x) x)

simplify137.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

80.0ms
(* -1 (/ (- 1 (exp x)) x))
41.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
13.0ms
(/ (- (exp x) 1) x)

prune157.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

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)))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

2.0ms
(/ (expm1 x) x)
0.0ms
(expm1 (/ (expm1 x) x))
0.0ms
(log1p (expm1 (/ (expm1 x) x)))

series349.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

292.0ms
(log1p (expm1 (/ (expm1 x) x)))
30.0ms
(expm1 (/ (expm1 x) x))
27.0ms
(/ (expm1 x) x)

simplify1.2s

Counts
14 → 47
Calls

14 calls. Slowest were:

431.0ms
(- (+ (* 7/24 (* (pow x 2) E)) (+ E (* 1/2 (* x E)))) 1)
254.0ms
(- (+ (/ (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)))))
208.0ms
(- (+ (/ (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)))))

prune421.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

0.9b
(* (sqrt (expm1 (/ (expm1 x) x))) (sqrt (expm1 (/ (expm1 x) x))))
0.7b
(expm1 (/ (expm1 x) x))
0.7b
(expm1 (/ (expm1 x) x))
0.0b
(sqrt (expm1 (/ (expm1 x) x)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (expm1 (/ (expm1 x) x))) (sqrt (expm1 (/ (expm1 x) x))))
1.0ms
(sqrt (expm1 (/ (expm1 x) x)))
0.0ms
(expm1 (/ (expm1 x) x))

series172.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

60.0ms
(sqrt (expm1 (/ (expm1 x) x)))
41.0ms
(* (sqrt (expm1 (/ (expm1 x) x))) (sqrt (expm1 (/ (expm1 x) x))))
37.0ms
(expm1 (/ (expm1 x) x))
34.0ms
(expm1 (/ (expm1 x) x))

simplify3.3s

Counts
40 → 80
Calls

40 calls. Slowest were:

651.0ms
(- (+ (/ (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)))))
411.0ms
(- (+ (* 7/24 (* (pow x 2) E)) (+ E (* 1/2 (* x E)))) 1)
388.0ms
(- (+ (* 7/24 (* (pow x 2) E)) (+ E (* 1/2 (* x E)))) 1)

prune984.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize16.0ms

Local error

Found 2 expressions with local error:

0.1b
(fma x 1/6 1/2)
0.0b
(fma x (fma x 1/6 1/2) 1)

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
2 → 20
Calls

2 calls. Slowest were:

0.0ms
(fma x 1/6 1/2)
0.0ms
(fma x (fma x 1/6 1/2) 1)

series45.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

23.0ms
(fma x 1/6 1/2)
21.0ms
(fma x (fma x 1/6 1/2) 1)

simplify185.0ms

Counts
6 → 26
Calls

6 calls. Slowest were:

65.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
65.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
34.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))

prune327.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes36.0ms

Accuracy

0% (0.0b remaining)

Error of 0.1b against oracle of 0.1b and baseline of 0.1b

bsearch1.0ms