Average Error: 4.1 → 0.2
Time: 25.3s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{e^{x \cdot 3} + 1}{\frac{\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right) - \left(1 - e^{x}\right) \cdot \left(1 - e^{x}\right)}{e^{x} \cdot e^{x} - \left(1 - e^{x}\right)}}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.1

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

  2. Simplified0.0

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

  6. Applied pow-exp0.2

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

  8. Applied flip-+0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 24.8s)Debug log

sample389.0ms

Algorithm
intervals

simplify113.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

112.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

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

series57.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

43.0ms
(sqrt (+ (exp x) 1))
14.0ms
(+ (exp x) 1)

simplify301.0ms

Counts
12 → 30
Calls

12 calls. Slowest were:

179.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
42.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
37.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune256.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize42.0ms

Local error

Found 4 expressions with local error:

4.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.0b
(* (exp x) (exp x))
0.0b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

7.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
5.0ms
(- (* 1 1) (* (exp x) 1))
4.0ms
(* (exp x) (exp x))

series70.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

23.0ms
(pow (exp x) 3)
21.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
14.0ms
(* (exp x) (exp x))
12.0ms
(- (* 1 1) (* (exp x) 1))

simplify722.0ms

Counts
39 → 79
Calls

39 calls. Slowest were:

181.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
120.0ms
(* (exp (* (exp x) (exp x))) (exp (- (* 1 1) (* (exp x) 1))))
104.0ms
(* (exp (* (exp x) (exp x))) (/ (exp (* 1 1)) (exp (* (exp x) 1))))

prune959.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

4.5b
(- (* 1 1) (* (exp x) 1))
0.0b
(* (exp x) (exp x))
0.0b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
0.0b
(/ (+ (exp (* x 3)) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
Calls

4 calls. Slowest were:

35.0ms
(/ (+ (exp (* x 3)) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
7.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
4.0ms
(* (exp x) (exp x))

series106.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

45.0ms
(/ (+ (exp (* x 3)) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
27.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
19.0ms
(- (* 1 1) (* (exp x) 1))
15.0ms
(* (exp x) (exp x))

simplify7.1s

Counts
85 → 107
Calls

85 calls. Slowest were:

604.0ms
(+ (* (* (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)))))
354.0ms
(/ (+ (* (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))))))
339.0ms
(/ (+ (* (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))))))

prune1.4s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize35.0ms

Local error

Found 4 expressions with local error:

4.5b
(- (* 1 1) (* (exp x) 1))
4.5b
(- (* 1 1) (* (exp x) 1))
4.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(* (* (exp x) (exp x)) (* (exp x) (exp x)))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

30.0ms
(* (* (exp x) (exp x)) (* (exp x) (exp x)))
4.0ms
(- (* 1 1) (* (exp x) 1))
4.0ms
(- (* 1 1) (* (exp x) 1))

series72.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

21.0ms
(* (* (exp x) (exp x)) (* (exp x) (exp x)))
17.0ms
(- (* 1 1) (* (exp x) 1))
17.0ms
(- (* 1 1) (* (exp x) 1))
17.0ms
(- (* 1 1) (* (exp x) 1))

simplify2.0s

Counts
69 → 94
Calls

69 calls. Slowest were:

323.0ms
(* (* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x))) (* (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (exp x) (exp x))))
300.0ms
(* (* (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (exp x) (exp x))) (* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x))))
258.0ms
(* (* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x))) (* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x))))

prune1.4s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes21.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch3.0ms

end0.0ms

sample9.6s

Algorithm
intervals