Average Error: 4.2 → 0.1
Time: 23.6s
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 2019016 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 23.3s)Debug log

sample929.0ms

Algorithm
intervals
Results
814×(pre true 80)
372×(body nan 80)
244×(body real 80)
183×(body exit 10240)
15×(body real 160)

simplify47.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
47.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

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
4.0ms
(sqrt (+ (exp x) 1))
2.0ms
(+ (exp x) 1)

series35.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
28.0ms
(sqrt (+ (exp x) 1))
7.0ms
(+ (exp x) 1)

simplify197.0ms

Counts
12 → 30
Calls
12 calls:
Slowest
114.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
28.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
26.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
8.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
4.0ms
(+ (exp x) 1)

prune257.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize28.0ms

Local error

Found 4 expressions with local error:

3.7b
(- (* 1 1) (* (exp x) 1))
0.3b
(/ (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
0.3b
(sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
0.1b
(pow (exp x) 3)

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
45×sqrt-prod
36×times-frac
36×add-sqr-sqrt
35×add-cube-cbrt
35×*-un-lft-identity
7×add-cbrt-cube
6×add-log-exp
6×associate-/l*
6×add-exp-log
6×sqrt-div
6×associate-/r*
5×pow1
3×flip-+
3×cube-prod
3×flip3-+
3×unpow-prod-down
2×associate-/l/
2×associate-/r/
1×rem-cube-cbrt
1×flip--
1×div-inv
1×pow-exp
1×div-exp
1×unpow3
1×diff-log
1×pow1/2
1×distribute-rgt-out--
1×flip3--
1×frac-2neg
1×pow-pow
1×sub-neg
1×clear-num
1×sqrt-undiv
1×rem-sqrt-square
1×cbrt-undiv
1×pow-to-exp
1×cube-mult
Counts
4 → 111
Calls
4 calls:
Slowest
13.0ms
(/ (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
8.0ms
(sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
2.0ms
(- (* 1 1) (* (exp x) 1))
1.0ms
(pow (exp x) 3)

series161.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
80.0ms
(/ (sqrt (+ 1 (pow (exp x) 3))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
50.0ms
(sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
19.0ms
(pow (exp x) 3)
12.0ms
(- (* 1 1) (* (exp x) 1))

simplify9.9s

Counts
119 → 123
Calls
119 calls:
Slowest
645.0ms
(sqrt (+ (* (* (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))))))
551.0ms
(sqrt (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1)))))
193.0ms
(sqrt (+ (pow (* (exp x) (exp x)) 3) (pow (- (* 1 1) (* (exp x) 1)) 3)))
191.0ms
(sqrt (- (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
158.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))

prune1.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes19.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample10.0s

Algorithm
intervals
Results
27061×(pre true 80)
12412×(body nan 80)
7431×(body real 80)
6508×(body exit 10240)
710×(body real 160)