Average Error: 40.5 → 0.5
Time: 8.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{1}{\frac{1}{e^{x}} \cdot (e^{x} - 1)^*}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.5
Target40.1
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.5

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

  2. Simplified0.5

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

  4. Applied clear-num0.5

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

  6. Applied div-inv0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))

Details

Time bar (total: 7.4s)Debug log

sample95.0ms

Algorithm
intervals
Results
32.0ms81×body1280valid
30.0ms85×body10240exit
14.0ms51×body640valid
5.0ms26×body320valid
4.0ms89×body80valid
1.0ms10×body160valid

simplify3.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
3.0ms
(/ (exp x) (- (exp x) 1))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize15.0ms

Local error

Found 1 expressions with local error:

1.0b
(/ (exp x) (expm1 x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×times-frac
9×*-un-lft-identity
9×add-sqr-sqrt
3×associate-/l*
3×add-cbrt-cube
3×associate-/r*
2×add-exp-log
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
Counts
1 → 29
Calls
1 calls:
Slowest
4.0ms
(/ (exp x) (expm1 x))

series22.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
22.0ms
(/ (exp x) (expm1 x))

simplify283.0ms

Counts
28 → 32
Calls
28 calls:
Slowest
150.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
29.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
19.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) 1)
12.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (sqrt (expm1 x)))
12.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune255.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.9b

localize11.0ms

Local error

Found 2 expressions with local error:

1.0b
(/ (expm1 x) (exp x))
0.0b
(/ 1 (/ (expm1 x) (exp x)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
16×associate-/r*
5×add-exp-log
4×add-cbrt-cube
3×associate-/l*
3×div-inv
3×pow1
2×add-log-exp
2×log1p-expm1-u
2×div-exp
2×frac-2neg
2×clear-num
2×rec-exp
2×expm1-log1p-u
1×expm1-udef
1×inv-pow
1×pow-flip
1×associate-/r/
1×div-sub
1×cbrt-undiv
Counts
2 → 60
Calls
2 calls:
Slowest
7.0ms
(/ (expm1 x) (exp x))
3.0ms
(/ 1 (/ (expm1 x) (exp x)))

series64.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
37.0ms
(/ (expm1 x) (exp x))
28.0ms
(/ 1 (/ (expm1 x) (exp x)))

simplify584.0ms

Counts
47 → 66
Calls
47 calls:
Slowest
273.0ms
(- (+ x (* 1/6 (pow x 3))) (* 1/2 (pow x 2)))
153.0ms
(/ (* (* (expm1 x) (expm1 x)) (expm1 x)) (* (* (exp x) (exp x)) (exp x)))
29.0ms
(/ (* (cbrt (expm1 x)) (cbrt (expm1 x))) (* (cbrt (exp x)) (cbrt (exp x))))
14.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
9.0ms
(/ (* (cbrt (expm1 x)) (cbrt (expm1 x))) 1)

prune571.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 2 expressions with local error:

1.0b
(* (expm1 x) (/ 1 (exp x)))
0.0b
(/ 1 (* (expm1 x) (/ 1 (exp x))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
7×pow1
5×rec-exp
4×add-cube-cbrt
4×prod-exp
4×associate-*r*
4×add-cbrt-cube
4×*-un-lft-identity
4×add-sqr-sqrt
3×associate-*l*
2×add-log-exp
2×pow-flip
2×div-inv
2×log1p-expm1-u
2×un-div-inv
2×associate-/r/
2×associate-*r/
2×pow-prod-down
2×expm1-log1p-u
1×inv-pow
1×cbrt-unprod
1×*-commutative
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
2 → 44
Calls
2 calls:
Slowest
8.0ms
(* (expm1 x) (/ 1 (exp x)))
6.0ms
(/ 1 (* (expm1 x) (/ 1 (exp x))))

series70.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
36.0ms
(/ 1 (* (expm1 x) (/ 1 (exp x))))
34.0ms
(* (expm1 x) (/ 1 (exp x)))

simplify780.0ms

Counts
23 → 50
Calls
23 calls:
Slowest
387.0ms
(* (* (* (expm1 x) (expm1 x)) (expm1 x)) (* (* (/ 1 (exp x)) (/ 1 (exp x))) (/ 1 (exp x))))
294.0ms
(- (+ x (* 1/6 (pow x 3))) (* 1/2 (pow x 2)))
20.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
14.0ms
(* (expm1 x) (/ 1 (exp x)))
8.0ms
(/ (- (exp x) 1) (exp x))

prune407.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize1.0ms

Local error

Found 1 expressions with local error:

1.0b
(/ (exp x) (expm1 x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×times-frac
9×*-un-lft-identity
9×add-sqr-sqrt
3×associate-/l*
3×add-cbrt-cube
3×associate-/r*
2×add-exp-log
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
Counts
1 → 29
Calls
1 calls:
Slowest
6.0ms
(/ (exp x) (expm1 x))

series35.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
35.0ms
(/ (exp x) (expm1 x))

simplify257.0ms

Counts
28 → 32
Calls
28 calls:
Slowest
120.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
28.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
21.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) 1)
18.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
12.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (sqrt (expm1 x)))

prune221.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes73.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample3.6s

Algorithm
intervals
Results
1.4s2600×body10240exit
1.1s2250×body1280valid
544.0ms1685×body640valid
172.0ms883×body320valid
166.0ms2806×body80valid
47.0ms379×body160valid