Average Error: 39.8 → 0.4
Time: 20.1s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt{e^{x}}}{\frac{(e^{x} - 1)^*}{\sqrt{e^{x}}}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.4
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.8

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

  2. Initial simplification0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2018365 +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: 13.2s)Debug log

start105.0ms

Algorithm
intervals

setup10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize8.0ms

Local error

Found 1 expressions with local error:

0.7b
(/ (exp x) (expm1 x))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

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

series23.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

23.0ms
(/ (exp x) (expm1 x))

simplify248.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

148.0ms
(/ (* (* (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))))
20.0ms
(+ (* 1/12 x) (+ (/ 1 x) 1/2))

prune318.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.7b

localize7.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (sqrt (exp x)) (/ (expm1 x) (sqrt (exp x))))
0.6b
(/ (expm1 x) (sqrt (exp x)))
0.0b
(sqrt (exp x))
0.0b
(sqrt (exp x))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 247
Calls

4 calls. Slowest were:

6.0ms
(/ (sqrt (exp x)) (/ (expm1 x) (sqrt (exp x))))
3.0ms
(/ (expm1 x) (sqrt (exp x)))
1.0ms
(sqrt (exp x))

series162.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(/ (expm1 x) (sqrt (exp x)))
38.0ms
(/ (sqrt (exp x)) (/ (expm1 x) (sqrt (exp x))))
20.0ms
(sqrt (exp x))
18.0ms
(sqrt (exp x))

simplify7.7s

Counts
363 → 259
Calls

363 calls. Slowest were:

496.0ms
(/ (sqrt 1) (/ (* (cbrt (expm1 x)) (cbrt (expm1 x))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))))
417.0ms
(/ (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x)))) (/ 1 (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))))
378.0ms
(/ (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (/ (expm1 x) (sqrt (exp x))) (/ (expm1 x) (sqrt (exp x)))) (/ (expm1 x) (sqrt (exp x)))))

prune2.9s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.7b

localize21.0ms

Local error

Found 2 expressions with local error:

0.1b
(fma 1/12 x 1/2)
0.0b
(+ (fma 1/12 x 1/2) (/ 1 x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

6.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))
0.0ms
(fma 1/12 x 1/2)

series35.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

23.0ms
(fma 1/12 x 1/2)
12.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))

simplify136.0ms

Counts
12 → 35
Calls

12 calls. Slowest were:

38.0ms
(* (exp (fma 1/12 x 1/2)) (exp (/ 1 x)))
19.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))
15.0ms
(+ (fma 1/12 x 1/2) (/ 1 x))

prune266.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.7b

localize9.0ms

Local error

Found 4 expressions with local error:

0.6b
(* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2)))
0.1b
(fma 1/12 x 1/2)
0.1b
(fma 1/12 x 1/2)
0.0b
(+ (* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2))) (/ 1 x))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

7.0ms
(+ (* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2))) (/ 1 x))
7.0ms
(* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2)))
0.0ms
(fma 1/12 x 1/2)

series55.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

19.0ms
(fma 1/12 x 1/2)
14.0ms
(* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2)))
14.0ms
(fma 1/12 x 1/2)
8.0ms
(+ (* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2))) (/ 1 x))

simplify343.0ms

Counts
35 → 78
Calls

35 calls. Slowest were:

143.0ms
(* (* (* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2))) (sqrt (fma 1/12 x 1/2))) (* (* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2))) (sqrt (fma 1/12 x 1/2))))
33.0ms
(* (exp (* (sqrt (fma 1/12 x 1/2)) (sqrt (fma 1/12 x 1/2)))) (exp (/ 1 x)))
11.0ms
(+ (log (sqrt (fma 1/12 x 1/2))) (log (sqrt (fma 1/12 x 1/2))))

prune849.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.7b

regimes52.0ms

Accuracy

0% (0.1b remaining)

Error of 0.4b against oracle of 0.3b and baseline of 0.4b

bsearch1.0ms