Average Error: 4.5 → 0.1
Time: 23.5s
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 2019021 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 23.2s)Debug log

sample347.0ms

Algorithm
intervals
Results
137.0ms186×body10240exit
31.0ms369×body80nan
22.0ms236×body80valid
5.0ms24×body160valid

simplify80.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
80.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune22.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize20.0ms

Local error

Found 2 expressions with local error:

0.2b
(sqrt (+ (exp x) 1))
0.0b
(+ (exp x) 1)

rewrite14.0ms

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
7.0ms
(sqrt (+ (exp x) 1))
4.0ms
(+ (exp x) 1)

series49.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
38.0ms
(sqrt (+ (exp x) 1))
10.0ms
(+ (exp x) 1)

simplify257.0ms

Counts
12 → 30
Calls
12 calls:
Slowest
166.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
34.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
21.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
10.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
5.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune235.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize41.0ms

Local error

Found 4 expressions with local error:

5.1b
(- (* 1 1) (* (exp x) 1))
0.2b
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
0.2b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
0.2b
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))

rewrite60.0ms

Algorithm
rewrite-expression-head
Rules
46×*-un-lft-identity
37×times-frac
28×add-sqr-sqrt
25×add-cube-cbrt
24×sqrt-prod
14×distribute-lft-out
11×add-log-exp
7×sum-cubes
7×cube-mult
6×associate-/l*
6×add-exp-log
6×add-cbrt-cube
4×flip-+
4×associate-/r/
4×flip3-+
4×pow1
3×associate-/r*
2×associate-/l/
2×div-inv
2×diff-log
2×sub-neg
2×sum-log
2×associate-+r+
2×rem-sqrt-square
1×flip--
1×div-exp
1×sqrt-div
1×pow1/2
1×distribute-rgt-out--
1×flip3--
1×frac-2neg
1×associate-+r-
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
4 → 106
Calls
4 calls:
Slowest
24.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
19.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
7.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
5.0ms
(- (* 1 1) (* (exp x) 1))

series212.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
94.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
61.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
33.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
23.0ms
(- (* 1 1) (* (exp x) 1))

simplify8.9s

Counts
94 → 118
Calls
94 calls:
Slowest
622.0ms
(+ (* (* (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)))))
542.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1))))))
392.0ms
(sqrt (/ (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))) (* (cbrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))) (cbrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))))
348.0ms
(/ (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))) (* (cbrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))) (cbrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))))
215.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (pow (* (exp x) (exp x)) 3) (pow (- (* 1 1) (* (exp x) 1)) 3))))

prune1.4s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

localize37.0ms

Local error

Found 4 expressions with local error:

5.1b
(- (* 1 1) (* (exp x) 1))
5.1b
(- (* 1 1) (* (exp x) 1))
5.1b
(- (* 1 1) (* (exp x) 1))
5.1b
(- (* 1 1) (* (exp x) 1))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
12×add-log-exp
4×flip--
4×add-cube-cbrt
4×add-exp-log
4×diff-log
4×distribute-rgt-out--
4×add-cbrt-cube
4×flip3--
4×*-un-lft-identity
4×sub-neg
4×pow1
4×add-sqr-sqrt
Counts
4 → 52
Calls
4 calls:
Slowest
4.0ms
(- (* 1 1) (* (exp x) 1))
4.0ms
(- (* 1 1) (* (exp x) 1))
4.0ms
(- (* 1 1) (* (exp x) 1))
4.0ms
(- (* 1 1) (* (exp x) 1))

series67.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
19.0ms
(- (* 1 1) (* (exp x) 1))
19.0ms
(- (* 1 1) (* (exp x) 1))
18.0ms
(- (* 1 1) (* (exp x) 1))
11.0ms
(- (* 1 1) (* (exp x) 1))

simplify978.0ms

Counts
20 → 64
Calls
20 calls:
Slowest
225.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
195.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
192.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
172.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
43.0ms
(/ (exp (* 1 1)) (exp (* (exp x) 1)))

prune1.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.2b

regimes21.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.4s

Algorithm
intervals
Results
6.4s6874×body10240exit
1.9s13064×body80nan
777.0ms7397×body80valid
160.0ms746×body160valid