Average Error: 4.2 → 0.1
Time: 14.6s
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 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 14.3s)Debug log

sample672.0ms

Algorithm
intervals
Results
435.0ms235×body10240exit
58.0ms455×body80nan
56.0ms239×body80valid
6.0ms21×body160valid

simplify104.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
103.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
5×add-sqr-sqrt
4×add-cube-cbrt
4×*-un-lft-identity
3×sqrt-prod
3×fma-def
2×add-log-exp
2×log1p-expm1-u
2×flip-+
2×add-exp-log
2×sqrt-div
2×add-cbrt-cube
2×flip3-+
2×pow1
2×expm1-log1p-u
1×pow1/2
1×rem-sqrt-square
1×+-commutative
Counts
2 → 31
Calls
2 calls:
Slowest
8.0ms
(+ (exp x) 1)
5.0ms
(sqrt (+ (exp x) 1))

series60.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
38.0ms
(sqrt (+ (exp x) 1))
21.0ms
(+ (exp x) 1)

simplify567.0ms

Counts
14 → 37
Calls
14 calls:
Slowest
325.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
105.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
90.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
14.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
4.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune318.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

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)))

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
28×sqrt-prod
28×sqrt-div
16×add-sqr-sqrt
14×add-cube-cbrt
14×*-un-lft-identity
9×flip-+
9×associate-*l*
9×associate-*r*
9×flip3-+
9×pow1
6×add-exp-log
6×add-cbrt-cube
6×fma-def
5×pow1/2
4×add-log-exp
4×log1p-expm1-u
4×frac-times
4×expm1-log1p-u
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 → 96
Calls
4 calls:
Slowest
26.0ms
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
7.0ms
(sqrt (sqrt (+ (exp x) 1)))
5.0ms
(+ (exp x) 1)
5.0ms
(+ (exp x) 1)

series174.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
122.0ms
(sqrt (sqrt (+ (exp x) 1)))
34.0ms
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
11.0ms
(+ (exp x) 1)
6.0ms
(+ (exp x) 1)

simplify1.3s

Counts
66 → 108
Calls
66 calls:
Slowest
335.0ms
(+ (* 1/8 (* x (pow 2 1/4))) (+ (* 5/128 (* (pow x 2) (pow 2 1/4))) (pow 2 1/4)))
127.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
104.0ms
(* (sqrt (sqrt (- (exp x) 1))) (sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
89.0ms
(sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
76.0ms
(* (sqrt (sqrt (- (* (exp x) (exp x)) (* 1 1)))) (sqrt (sqrt (+ (pow (exp x) 3) (pow 1 3)))))

prune1.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes24.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.8s

Algorithm
intervals
Results
6.4s6681×body10240exit
1.5s12774×body80nan
1.4s7430×body80valid
163.0ms693×body160valid