Average Error: 4.4 → 0.1
Time: 23.3s
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 4.4

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 23.0s)Debug log

sample662.0ms

Algorithm
intervals

simplify154.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

153.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 2 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

5.0ms
(sqrt (+ (exp x) 1))
4.0ms
(+ (exp x) 1)

series54.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

43.0ms
(sqrt (+ (exp x) 1))
11.0ms
(+ (exp x) 1)

simplify244.0ms

Counts
12 → 30
Calls

12 calls. Slowest were:

156.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
27.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
25.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune285.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize49.0ms

Local error

Found 4 expressions with local error:

5.2b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.0b
(/ (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
0.0b
(* (exp x) (exp x))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 117
Calls

4 calls. Slowest were:

13.0ms
(/ (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
3.0ms
(* (exp x) (exp x))
2.0ms
(- (* 1 1) (* (exp x) 1))

series162.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

119.0ms
(/ (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
21.0ms
(pow (exp x) 3)
12.0ms
(- (* 1 1) (* (exp x) 1))
10.0ms
(* (exp x) (exp x))

simplify8.7s

Counts
124 → 129
Calls

124 calls. Slowest were:

672.0ms
(sqrt (+ (* (* (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))))))
198.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
176.0ms
(/ (* (* (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ 1 (pow (exp x) 3)))) (sqrt (+ 1 (pow (exp x) 3)))) (* (* (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))

prune1.5s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

5.2b
(- (* 1 1) (* (exp x) 1))
0.2b
(sqrt (exp x))
0.2b
(sqrt (exp x))
0.1b
(pow (sqrt (exp x)) 3)

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

2.0ms
(- (* 1 1) (* (exp x) 1))
1.0ms
(pow (sqrt (exp x)) 3)
1.0ms
(sqrt (exp x))

series91.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

34.0ms
(pow (sqrt (exp x)) 3)
24.0ms
(sqrt (exp x))
20.0ms
(sqrt (exp x))
12.0ms
(- (* 1 1) (* (exp x) 1))

simplify425.0ms

Counts
38 → 75
Calls

38 calls. Slowest were:

173.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
40.0ms
(/ (exp (* 1 1)) (exp (* (exp x) 1)))
37.0ms
(+ (* 3/2 x) (+ (* 9/8 (pow x 2)) 1))

prune1.0s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes22.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.5s

Algorithm
intervals