Average Error: 40.3 → 0.0
Time: 4.0s
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

Original40.3
Target39.5
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 40.3

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

  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 2019022 +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: 3.7s)Debug log

sample78.0ms

Algorithm
intervals
Results
40.0ms82×body1280valid
13.0ms51×body640valid
10.0ms176×body80valid
5.0ms27×body320valid
1.0ms9×body160valid

simplify11.0ms

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

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

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
2.0ms
(/ (expm1 x) x)

series23.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
23.0ms
(/ (expm1 x) x)

simplify181.0ms

Counts
6 → 19
Calls
6 calls:
Slowest
110.0ms
(* -1 (/ (- 1 (exp x)) x))
59.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
10.0ms
(/ (- (exp x) 1) x)
1.0ms
(sqrt (expm1 x))
1.0ms
(* (cbrt (expm1 x)) (cbrt (expm1 x)))

prune107.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize5.0ms

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

series69.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
37.0ms
(/ 1 (/ x (expm1 x)))
32.0ms
(/ x (expm1 x))

simplify245.0ms

Counts
15 → 41
Calls
15 calls:
Slowest
95.0ms
(* -1 (/ (- 1 (exp x)) x))
66.0ms
(- (+ (* 1/12 (pow x 2)) 1) (* 1/2 x))
57.0ms
(+ (* 1/2 x) (+ (* 1/6 (pow x 2)) 1))
7.0ms
(/ (- (exp x) 1) x)
6.0ms
(/ x (- (exp x) 1))

prune257.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes18.0ms

Accuracy

0% (0.0b remaining)

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

bsearch0.0ms

end0.0ms

sample2.7s

Algorithm
intervals
Results
960.0ms2268×body1280valid
772.0ms5437×body80valid
523.0ms1694×body640valid
195.0ms889×body320valid
81.0ms383×body160valid