Average Error: 4.2 → 0.1
Time: 24.6s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{x} + 1}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.2

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]

  3. Final simplification0.1

    \[\leadsto \sqrt{e^{x} + 1}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 24.1s)Debug log

sample545.0ms

Algorithm
intervals

simplify104.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

103.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 2 expressions with local error:

0.0b
(+ (exp x) 1)
0.0b
(sqrt (+ (exp x) 1))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

8.0ms
(+ (exp x) 1)
6.0ms
(sqrt (+ (exp x) 1))

series71.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

46.0ms
(sqrt (+ (exp x) 1))
24.0ms
(+ (exp x) 1)

simplify253.0ms

Counts
12 → 30
Calls

12 calls. Slowest were:

140.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
46.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
23.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune270.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize35.0ms

Local error

Found 4 expressions with local error:

5.8b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.1b
(* (exp x) (exp x))
0.0b
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
Calls

4 calls. Slowest were:

23.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
8.0ms
(- (* 1 1) (* (exp x) 1))
4.0ms
(* (exp x) (exp x))

series135.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

83.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
24.0ms
(pow (exp x) 3)
15.0ms
(* (exp x) (exp x))
13.0ms
(- (* 1 1) (* (exp x) 1))

simplify5.7s

Counts
82 → 107
Calls

82 calls. Slowest were:

534.0ms
(+ (* (* (exp x) (exp x)) (* (exp x) (exp x))) (- (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1))) (* (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
368.0ms
(/ (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))) (* (cbrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))) (cbrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
220.0ms
(/ (* (* (+ (pow (exp x) 3) (pow 1 3)) (+ (pow (exp x) 3) (pow 1 3))) (+ (pow (exp x) 3) (pow 1 3))) (* (* (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))

prune1.4s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize51.0ms

Local error

Found 4 expressions with local error:

5.8b
(- 1 (exp x))
5.8b
(- 1 (exp x))
5.8b
(- 1 (exp x))
0.1b
(/ (+ (exp (+ x x)) (- 1 (exp x))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

41.0ms
(/ (+ (exp (+ x x)) (- 1 (exp x))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))
2.0ms
(- 1 (exp x))
2.0ms
(- 1 (exp x))

series75.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

27.0ms
(/ (+ (exp (+ x x)) (- 1 (exp x))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))
18.0ms
(- 1 (exp x))
18.0ms
(- 1 (exp x))
12.0ms
(- 1 (exp x))

simplify4.7s

Counts
86 → 102
Calls

86 calls. Slowest were:

500.0ms
(- (* (exp (+ x x)) (exp (+ x x))) (* (- 1 (exp x)) (- 1 (exp x))))
461.0ms
(sqrt (+ (* (exp (+ x x)) (exp (+ x x))) (- (* (- 1 (exp x)) (- 1 (exp x))) (* (exp (+ x x)) (- 1 (exp x))))))
185.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))

prune1.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes27.0ms

Accuracy

0% (0.0b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch1.0ms

end0.0ms

sample9.5s

Algorithm
intervals