Average Error: 4.2 → 0.1
Time: 13.8s
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 4.2

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019022 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 13.5s)Debug log

sample333.0ms

Algorithm
intervals
Results
136.0ms235×body10240exit
34.0ms455×body80nan
17.0ms239×body80valid
3.0ms21×body160valid

simplify49.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
48.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune11.0ms

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))

rewrite348.0ms

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
341.0ms
(+ (exp x) 1)
5.0ms
(sqrt (+ (exp x) 1))

series55.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
31.0ms
(sqrt (+ (exp x) 1))
23.0ms
(+ (exp x) 1)

simplify234.0ms

Counts
12 → 30
Calls
12 calls:
Slowest
125.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
41.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
24.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
16.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
5.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune282.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
0.0b
(+ (exp x) 1)
0.0b
(+ (exp x) 1)
0.0b
(sqrt (sqrt (+ (exp x) 1)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
28×sqrt-prod
28×sqrt-div
14×add-sqr-sqrt
12×add-cube-cbrt
12×*-un-lft-identity
9×flip-+
9×associate-*l*
9×associate-*r*
9×flip3-+
9×pow1
6×add-exp-log
6×add-cbrt-cube
5×pow1/2
4×add-log-exp
4×frac-times
2×pow-prod-up
2×associate-*r/
2×associate-*l/
2×pow-prod-down
2×rem-sqrt-square
2×+-commutative
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×pow2
Counts
4 → 82
Calls
4 calls:
Slowest
12.0ms
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
6.0ms
(sqrt (sqrt (+ (exp x) 1)))
5.0ms
(+ (exp x) 1)
2.0ms
(+ (exp x) 1)

series184.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
131.0ms
(sqrt (sqrt (+ (exp x) 1)))
31.0ms
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
11.0ms
(+ (exp x) 1)
10.0ms
(+ (exp x) 1)

simplify1.2s

Counts
62 → 94
Calls
62 calls:
Slowest
226.0ms
(+ (* 1/8 (* x (pow 2 1/4))) (+ (* 5/128 (* (pow x 2) (pow 2 1/4))) (pow 2 1/4)))
122.0ms
(* (sqrt (sqrt (- (exp x) 1))) (sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
117.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
113.0ms
(* (sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (sqrt (- (exp x) 1))))
82.0ms
(sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))

prune946.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes31.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample9.7s

Algorithm
intervals
Results
6.5s6681×body10240exit
1.4s7430×body80valid
1.3s12774×body80nan
186.0ms693×body160valid