Average Error: 40.9 → 0.4
Time: 9.1s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt{e^{x}}}{(e^{x} - 1)^*} \cdot \sqrt{e^{x}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.9
Target40.5
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.9

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

  2. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied times-frac0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019010 +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: 8.7s)Debug log

sample136.0ms

Algorithm
intervals

simplify3.0ms

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize7.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ (exp x) (expm1 x))

rewrite3.0ms

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
3.0ms
(/ (exp x) (expm1 x))

series22.0ms

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

simplify277.0ms

Counts
28 → 32
Calls
28 calls:
Slowest
153.0ms
(/ (* (* (exp x) (exp x)) (exp x)) (* (* (expm1 x) (expm1 x)) (expm1 x)))
26.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (* (cbrt (expm1 x)) (cbrt (expm1 x))))
21.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))
10.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) (sqrt (expm1 x)))
9.0ms
(/ (* (cbrt (exp x)) (cbrt (exp x))) 1)

prune283.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize7.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (sqrt (exp x)) (expm1 x))
0.2b
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
0.0b
(sqrt (exp x))
0.0b
(sqrt (exp x))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
38×add-sqr-sqrt
36×add-cube-cbrt
36×times-frac
36×*-un-lft-identity
30×sqrt-prod
22×associate-*r*
11×add-exp-log
11×add-cbrt-cube
6×associate-/l*
6×associate-*l*
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/r*
2×div-inv
2×cbrt-unprod
2×prod-exp
2×div-exp
2×pow1/2
2×rem-sqrt-square
2×cbrt-undiv
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×frac-2neg
1×clear-num
Counts
4 → 113
Calls
4 calls:
Slowest
13.0ms
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
6.0ms
(/ (sqrt (exp x)) (expm1 x))
1.0ms
(sqrt (exp x))
1.0ms
(sqrt (exp x))

series143.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
85.0ms
(/ (sqrt (exp x)) (expm1 x))
35.0ms
(* (sqrt (exp x)) (/ (sqrt (exp x)) (expm1 x)))
12.0ms
(sqrt (exp x))
12.0ms
(sqrt (exp x))

simplify1.3s

Counts
99 → 125
Calls
99 calls:
Slowest
364.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (/ (sqrt (exp x)) (expm1 x)) (/ (sqrt (exp x)) (expm1 x))) (/ (sqrt (exp x)) (expm1 x))))
270.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (expm1 x) (expm1 x)) (expm1 x))))
108.0ms
(/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (expm1 x) (expm1 x)) (expm1 x)))
84.0ms
(- (+ (* 7/5760 (pow x 3)) (/ 1 x)) (* 1/24 x))
62.0ms
(+ (* 1/2 x) (+ (* 1/8 (pow x 2)) 1))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

localize16.0ms

Local error

Found 1 expressions with local error:

0.1b
(fma 1/12 x 1/2)

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma 1/12 x 1/2)

series20.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
20.0ms
(fma 1/12 x 1/2)

simplify10.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
3.0ms
(+ (* 1/12 x) 1/2)
3.0ms
(+ (* 1/12 x) 1/2)
3.0ms
(+ (* 1/12 x) 1/2)

prune109.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

localize9.0ms

Local error

Found 3 expressions with local error:

0.1b
(fma 1/12 x 1/2)
0.0b
(expm1 (log1p (fma 1/12 x 1/2)))
0.0b
(log1p (fma 1/12 x 1/2))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
4×expm1-log1p-u
3×add-log-exp
3×log1p-expm1-u
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
3×add-sqr-sqrt
1×expm1-udef
1×fma-udef
1×log1p-expm1
1×expm1-log1p
1×log1p-udef
Counts
3 → 32
Calls
3 calls:
Slowest
0.0ms
(log1p (fma 1/12 x 1/2))
0.0ms
(fma 1/12 x 1/2)
0.0ms
(expm1 (log1p (fma 1/12 x 1/2)))

series427.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
378.0ms
(expm1 (log1p (fma 1/12 x 1/2)))
33.0ms
(log1p (fma 1/12 x 1/2))
15.0ms
(fma 1/12 x 1/2)

simplify1.3s

Counts
10 → 41
Calls
10 calls:
Slowest
542.0ms
(- (+ (log -1/12) (* 18 (/ 1 x))) (+ (log (/ -1 x)) (* 162 (/ 1 (pow x 2)))))
336.0ms
(- (+ (* 18 (/ 1 x)) (log 1/12)) (+ (log (/ 1 x)) (* 162 (/ 1 (pow x 2)))))
181.0ms
(- (+ (* 18 (/ (exp (- (log -1/12) (log (/ -1 x)))) x)) (exp (- (log -1/12) (log (/ -1 x))))) 1)
118.0ms
(- (+ (* 18 (/ (exp (- (log 1/12) (log (/ 1 x)))) x)) (exp (- (log 1/12) (log (/ 1 x))))) 1)
98.0ms
(- (+ (* 1/18 x) (log 3/2)) (* 1/648 (pow x 2)))

prune329.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

regimes20.0ms

Accuracy

0% (0.1b remaining)

Error of 0.4b against oracle of 0.4b and baseline of 0.4b

bsearch1.0ms

end0.0ms

sample3.2s

Algorithm
intervals