Average Error: 39.3 → 0.0
Time: 7.3s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[(e^{\log_* (1 + \frac{1}{\frac{x}{(e^{x} - 1)^*}})} - 1)^*\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.3
Target38.3
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 39.3

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

  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. Using strategy rm

  7. Applied expm1-log1p-u0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019016 +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: 6.8s)Debug log

sample59.0ms

Algorithm
intervals
Results
316×(pre true 80)
168×(body real 80)
64×(body real 1280)
46×(body real 640)
26×(body real 320)
12×(body real 160)

simplify12.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
11.0ms
(/ (- (exp x) 1) x)

prune8.0ms

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
Rules
3×associate-/l*
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×expm1-udef
1×div-inv
1×log1p-expm1-u
1×add-exp-log
1×div-sub
1×add-cbrt-cube
1×frac-2neg
1×clear-num
1×pow1
1×expm1-log1p-u
Counts
1 → 16
Calls
1 calls:
Slowest
3.0ms
(/ (expm1 x) x)

series32.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
32.0ms
(/ (expm1 x) x)

simplify158.0ms

Counts
6 → 19
Calls
6 calls:
Slowest
91.0ms
(* -1 (/ (- 1 (exp x)) x))
56.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
7.0ms
(/ (- (exp x) 1) x)
2.0ms
(* (cbrt (expm1 x)) (cbrt (expm1 x)))
1.0ms
(/ (exp x) x)

prune122.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
7×associate-/r*
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×div-inv
3×add-exp-log
3×pow1
2×add-log-exp
2×log1p-expm1-u
2×add-cbrt-cube
2×frac-2neg
2×clear-num
2×expm1-log1p-u
1×inv-pow
1×pow-flip
1×associate-/r/
1×rec-exp
Counts
2 → 35
Calls
2 calls:
Slowest
2.0ms
(/ 1 (/ x (expm1 x)))
1.0ms
(/ x (expm1 x))

series54.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
29.0ms
(/ 1 (/ x (expm1 x)))
25.0ms
(/ x (expm1 x))

simplify278.0ms

Counts
15 → 41
Calls
15 calls:
Slowest
98.0ms
(* -1 (/ (- 1 (exp x)) x))
78.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
64.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
14.0ms
(/ (- (exp x) 1) x)
6.0ms
(/ x (- (exp x) 1))

prune273.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.0b
(expm1 (log1p (/ 1 (/ x (expm1 x)))))
0.0b
(/ 1 (/ x (expm1 x)))
0.0b
(/ x (expm1 x))
0.0b
(log1p (/ 1 (/ x (expm1 x))))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
7×associate-/r*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×add-exp-log
5×pow1
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
3×div-inv
2×frac-2neg
2×clear-num
1×expm1-udef
1×inv-pow
1×pow-flip
1×associate-/r/
1×log1p-expm1
1×expm1-log1p
1×log1p-udef
1×rec-exp
Counts
4 → 57
Calls
4 calls:
Slowest
1.0ms
(/ 1 (/ x (expm1 x)))
1.0ms
(/ x (expm1 x))
0.0ms
(log1p (/ 1 (/ x (expm1 x))))
0.0ms
(expm1 (log1p (/ 1 (/ x (expm1 x)))))

series399.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
293.0ms
(expm1 (log1p (/ 1 (/ x (expm1 x)))))
48.0ms
(log1p (/ 1 (/ x (expm1 x))))
32.0ms
(/ 1 (/ x (expm1 x)))
26.0ms
(/ x (expm1 x))

simplify1.0s

Counts
22 → 69
Calls
22 calls:
Slowest
268.0ms
(- (+ (/ (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))))))
209.0ms
(- (+ (/ (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))))))
127.0ms
(* -1 (/ (- 1 (exp x)) x))
98.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
95.0ms
(* -1 (/ (- 1 (exp x)) x))

prune705.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 1 expressions with local error:

0.1b
(* (/ 1 x) (expm1 x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
4×associate-*l*
3×add-cube-cbrt
3×add-exp-log
3×associate-*r*
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
3×add-sqr-sqrt
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×expm1-log1p-u
Counts
1 → 21
Calls
1 calls:
Slowest
4.0ms
(* (/ 1 x) (expm1 x))

series26.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
26.0ms
(* (/ 1 x) (expm1 x))

simplify646.0ms

Counts
12 → 24
Calls
12 calls:
Slowest
432.0ms
(* (* (* (/ 1 x) (/ 1 x)) (/ 1 x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
127.0ms
(* -1 (/ (- 1 (exp x)) x))
37.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
18.0ms
(+ (log (/ 1 x)) (log (expm1 x)))
13.0ms
(/ (- (exp x) 1) x)

prune146.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes16.0ms

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
Results
10628×(pre true 80)
5514×(body real 80)
2240×(body real 1280)
1669×(body real 640)
789×(body real 320)
416×(body real 160)