Average Error: 4.2 → 0.2
Time: 2.5m
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]

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. Using strategy rm

  4. Applied flip3-+0.2

    \[\leadsto \sqrt{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}\]

  5. Final simplification0.2

    \[\leadsto \sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 2.5m)Debug log

sample516.0ms

Algorithm
intervals

simplify80.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

79.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize12.0ms

Local error

Found 2 expressions with local error:

0.3b
(sqrt (+ (exp x) 1))
0.0b
(+ (exp x) 1)

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

5.0ms
(sqrt (+ (exp x) 1))
2.0ms
(+ (exp x) 1)

series59.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

47.0ms
(sqrt (+ (exp x) 1))
12.0ms
(+ (exp x) 1)

simplify169.0ms

Counts
12 → 30
Calls

12 calls. Slowest were:

97.0ms
(- (+ (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)))
19.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune228.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.2b

localize33.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (* 1 1) (* (exp x) 1))
0.3b
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
0.2b
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
0.2b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

rewrite87.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

45.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
23.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
14.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

series235.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

120.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
76.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
19.0ms
(- (* 1 1) (* (exp x) 1))
19.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

simplify8.5s

Counts
94 → 118
Calls

94 calls. Slowest were:

590.0ms
(+ (* (* (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)))))
521.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1))))))
335.0ms
(/ (+ (* (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))))))

prune1.5s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize0.0ms

Local error

Found 2 expressions with local error:

0.3b
(sqrt (+ (exp x) 1))
0.0b
(+ (exp x) 1)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

3.0ms
(sqrt (+ (exp x) 1))
2.0ms
(+ (exp x) 1)

series27.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

21.0ms
(sqrt (+ (exp x) 1))
7.0ms
(+ (exp x) 1)

simplify224.0ms

Counts
12 → 30
Calls

12 calls. Slowest were:

108.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
51.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
31.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune253.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

localize54.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (* 1 1) (* (exp x) 1))
4.1b
(- 1 (exp x))
0.3b
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x))))))
0.2b
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))

rewrite149.0ms

Algorithm
rewrite-expression-head
Counts
4 → 700
Calls

4 calls. Slowest were:

57.0ms
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))
48.0ms
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x))))))
4.0ms
(- (* 1 1) (* (exp x) 1))

series463.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

324.0ms
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x))))))
101.0ms
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))
19.0ms
(- (* 1 1) (* (exp x) 1))
18.0ms
(- 1 (exp x))

simplify1.9m

Counts
965 → 712
Calls

965 calls. Slowest were:

926.0ms
(/ (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)))))) (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))
750.0ms
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1)))))) (sqrt (sqrt (+ (exp (+ x x)) (- 1 (exp x)))))))
745.0ms
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1)))))) (sqrt (sqrt (+ (exp (+ x x)) (- 1 (exp x))))))

prune13.5s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.2b

regimes56.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 0.2b

bsearch4.0ms

end0.0ms

sample9.4s

Algorithm
intervals