Average Error: 4.4 → 0.1
Time: 25.4s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[e^{\log \left(\sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.4

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  3. Using strategy rm

  4. Applied add-exp-log0.1

    \[\leadsto \color{blue}{e^{\log \left(\sqrt{e^{x} + 1}\right)}}\]
  5. Using strategy rm

  6. Applied flip3-+0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 25.0s)Debug log

sample502.0ms

Algorithm
intervals

simplify74.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

73.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 2 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

8.0ms
(sqrt (+ (exp x) 1))
5.0ms
(+ (exp x) 1)

series42.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

34.0ms
(sqrt (+ (exp x) 1))
7.0ms
(+ (exp x) 1)

simplify217.0ms

Counts
14 → 37
Calls

14 calls. Slowest were:

120.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
39.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
31.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune225.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize5.0ms

Local error

Found 4 expressions with local error:

0.2b
(log (sqrt (+ (exp x) 1)))
0.1b
(exp (log (sqrt (+ (exp x) 1))))
0.0b
(sqrt (+ (exp x) 1))
0.0b
(+ (exp x) 1)

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

4.0ms
(exp (log (sqrt (+ (exp x) 1))))
3.0ms
(sqrt (+ (exp x) 1))
3.0ms
(log (sqrt (+ (exp x) 1)))

series124.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(log (sqrt (+ (exp x) 1)))
18.0ms
(exp (log (sqrt (+ (exp x) 1))))
18.0ms
(sqrt (+ (exp x) 1))
7.0ms
(+ (exp x) 1)

simplify1.1s

Counts
43 → 86
Calls

43 calls. Slowest were:

310.0ms
(- (+ (* 1/4 (/ (pow x 2) (pow (sqrt 2) 2))) (+ (log (sqrt 2)) (* 1/2 (/ x (pow (sqrt 2) 2))))) (* 1/4 (/ (pow x 2) (pow (sqrt 2) 4))))
234.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
193.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))

prune766.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize46.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (* 1 1) (* (exp x) 1))
0.2b
(log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
0.1b
(pow (exp x) 3)
0.1b
(exp (log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))))

rewrite57.0ms

Algorithm
rewrite-expression-head
Counts
4 → 121
Calls

4 calls. Slowest were:

29.0ms
(exp (log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))))
18.0ms
(log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
5.0ms
(- (* 1 1) (* (exp x) 1))

series367.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

218.0ms
(log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
107.0ms
(exp (log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))))
23.0ms
(pow (exp x) 3)
19.0ms
(- (* 1 1) (* (exp x) 1))

simplify8.1s

Counts
88 → 133
Calls

88 calls. Slowest were:

830.0ms
(log (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)))))))
768.0ms
(exp (log (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))))))))
385.0ms
(log (sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (pow (* (exp x) (exp x)) 3) (pow (- (* 1 1) (* (exp x) 1)) 3)))))

prune2.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 3 expressions with local error:

0.1b
(log (+ (exp x) 1))
0.1b
(exp (* 1/2 (log (+ (exp x) 1))))
0.0b
(+ (exp x) 1)

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
3 → 48
Calls

3 calls. Slowest were:

9.0ms
(+ (exp x) 1)
9.0ms
(exp (* 1/2 (log (+ (exp x) 1))))
5.0ms
(log (+ (exp x) 1))

series238.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

148.0ms
(exp (* 1/2 (log (+ (exp x) 1))))
78.0ms
(log (+ (exp x) 1))
11.0ms
(+ (exp x) 1)

simplify921.0ms

Counts
25 → 57
Calls

25 calls. Slowest were:

367.0ms
(+ (pow 2 1/2) (+ (* 1/4 (* (sqrt 2) x)) (* 3/32 (* (sqrt 2) (pow x 2)))))
134.0ms
(exp (* 1/2 (log (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))))
107.0ms
(exp (* (log (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1)))) 1/2))

prune533.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes39.0ms

Accuracy

0% (0.1b remaining)

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

bsearch4.0ms

end0.0ms

sample9.5s

Algorithm
intervals