Average Error: 4.5 → 0.1
Time: 14.4s
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 4.5

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019010 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 14.1s)Debug log

sample458.0ms

Algorithm
intervals

simplify93.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
92.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

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
Rules
4×add-sqr-sqrt
3×sqrt-prod
3×add-cube-cbrt
3×*-un-lft-identity
2×add-log-exp
2×flip-+
2×add-exp-log
2×sqrt-div
2×add-cbrt-cube
2×flip3-+
2×pow1
1×pow1/2
1×rem-sqrt-square
1×+-commutative
Counts
2 → 24
Calls
2 calls:
Slowest
9.0ms
(sqrt (+ (exp x) 1))
5.0ms
(+ (exp x) 1)

series57.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
46.0ms
(sqrt (+ (exp x) 1))
11.0ms
(+ (exp x) 1)

simplify219.0ms

Counts
12 → 30
Calls
12 calls:
Slowest
126.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
32.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
20.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
17.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
5.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune257.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize45.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.1b
(* (exp x) (exp x))
0.0b
(+ 1 (pow (exp x) 3))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
10×pow1
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×add-cbrt-cube
6×add-log-exp
4×add-exp-log
3×associate-*l*
3×associate-*r*
3×cube-prod
3×unpow-prod-down
1×rem-cube-cbrt
1×flip--
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-exp
1×pow-prod-down
1×unpow3
1×diff-log
1×distribute-rgt-out--
1×flip3--
1×pow-plus
1×flip3-+
1×pow-pow
1×sub-neg
1×+-commutative
1×pow-to-exp
1×cube-mult
1×pow2
Counts
4 → 62
Calls
4 calls:
Slowest
3.0ms
(* (exp x) (exp x))
2.0ms
(- (* 1 1) (* (exp x) 1))
1.0ms
(+ 1 (pow (exp x) 3))
1.0ms
(pow (exp x) 3)

series67.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
29.0ms
(+ 1 (pow (exp x) 3))
15.0ms
(pow (exp x) 3)
13.0ms
(- (* 1 1) (* (exp x) 1))
11.0ms
(* (exp x) (exp x))

simplify475.0ms

Counts
34 → 74
Calls
34 calls:
Slowest
185.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
43.0ms
(/ (exp (* 1 1)) (exp (* (exp x) 1)))
43.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x)))
37.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))
35.0ms
(+ (* 3 x) (+ (* 9/2 (pow x 2)) 2))

prune875.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.1b
(* (sqrt (exp x)) (exp x))
0.0b
(+ 1 (pow (exp x) 3))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
8×pow1
7×add-cbrt-cube
6×add-log-exp
6×associate-*l*
5×add-exp-log
3×sqrt-prod
3×associate-*r*
3×cube-prod
3×unpow-prod-down
2×pow1/2
1×rem-cube-cbrt
1×flip--
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-exp
1×pow-prod-down
1×cube-unmult
1×unpow3
1×diff-log
1×distribute-rgt-out--
1×flip3--
1×pow-plus
1×flip3-+
1×pow-pow
1×sub-neg
1×+-commutative
1×pow-to-exp
1×cube-mult
Counts
4 → 65
Calls
4 calls:
Slowest
7.0ms
(* (sqrt (exp x)) (exp x))
4.0ms
(- (* 1 1) (* (exp x) 1))
2.0ms
(+ 1 (pow (exp x) 3))
2.0ms
(pow (exp x) 3)

series96.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
34.0ms
(* (sqrt (exp x)) (exp x))
22.0ms
(- (* 1 1) (* (exp x) 1))
21.0ms
(pow (exp x) 3)
19.0ms
(+ 1 (pow (exp x) 3))

simplify454.0ms

Counts
37 → 77
Calls
37 calls:
Slowest
162.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
43.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (exp x) (exp x)) (exp x)))
35.0ms
(+ (* 3 x) (+ (* 9/2 (pow x 2)) 2))
29.0ms
(pow (* (cbrt (exp x)) (cbrt (exp x))) 3)
27.0ms
(pow (* (cbrt (exp x)) (cbrt (exp x))) 3)

prune1.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes37.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.7s

Algorithm
intervals