Average Error: 4.4 → 0.1
Time: 14.7s
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 4.4

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019018 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 14.3s)Debug log

sample270.0ms

Algorithm
intervals
Results
150.0ms191×body10240exit
30.0ms409×body80nan
17.0ms234×body80valid
4.0ms25×body160valid

simplify68.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
68.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

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)

series42.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
28.0ms
(sqrt (+ (exp x) 1))
14.0ms
(+ (exp x) 1)

simplify180.0ms

Counts
12 → 30
Calls
12 calls:
Slowest
102.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
31.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
21.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
9.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
3.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune191.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

5.9b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.1b
(* (exp x) (exp x))
0.0b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
11×add-log-exp
10×pow1
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×add-cbrt-cube
4×add-exp-log
3×associate-*l*
3×associate-*r*
3×cube-prod
3×unpow-prod-down
2×diff-log
2×sub-neg
2×sum-log
2×associate-+r+
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×distribute-rgt-out--
1×flip3--
1×pow-plus
1×flip3-+
1×pow-pow
1×associate-+r-
1×+-commutative
1×pow-to-exp
1×cube-mult
1×pow2
Counts
4 → 67
Calls
4 calls:
Slowest
10.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
3.0ms
(* (exp x) (exp x))
2.0ms
(- (* 1 1) (* (exp x) 1))
1.0ms
(pow (exp x) 3)

series56.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
19.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
15.0ms
(pow (exp x) 3)
12.0ms
(- (* 1 1) (* (exp x) 1))
10.0ms
(* (exp x) (exp x))

simplify915.0ms

Counts
39 → 79
Calls
39 calls:
Slowest
213.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
127.0ms
(* (exp (* (exp x) (exp x))) (exp (- (* 1 1) (* (exp x) 1))))
85.0ms
(* (exp (* (exp x) (exp x))) (/ (exp (* 1 1)) (exp (* (exp x) 1))))
74.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))
64.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x)))

prune952.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
28×sqrt-div
24×sqrt-prod
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×pow1/2
6×add-cbrt-cube
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
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×+-commutative
1×pow2
Counts
4 → 82
Calls
4 calls:
Slowest
22.0ms
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
8.0ms
(sqrt (+ (exp x) 1))
5.0ms
(sqrt (+ (exp x) 1))
5.0ms
(+ (exp x) 1)

series101.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
35.0ms
(sqrt (+ (exp x) 1))
33.0ms
(* (sqrt (sqrt (+ (exp x) 1))) (sqrt (sqrt (+ (exp x) 1))))
23.0ms
(sqrt (+ (exp x) 1))
10.0ms
(+ (exp x) 1)

simplify1.2s

Counts
64 → 94
Calls
64 calls:
Slowest
161.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
146.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
122.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
100.0ms
(* (sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (sqrt (- (exp x) 1))))
89.0ms
(* (sqrt (sqrt (- (exp x) 1))) (sqrt (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))

prune1.0s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes25.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.1s

Algorithm
intervals
Results
5.9s6603×body10240exit
1.8s12713×body80nan
956.0ms7375×body80valid
255.0ms744×body160valid