Average Error: 4.5 → 0.1
Time: 36.1s
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 1137770271 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 35.5s)Debug log

sample518.0ms

Algorithm
intervals
Results
222×270.0ms(body exit 10240)
406×69.0ms(body nan 80)
889×50.0ms(pre true 80)
238×42.0ms(body real 80)
23×9.0ms(body real 160)

simplify128.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
127.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 2 expressions with local error:

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

rewrite28.0ms

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
9.0ms
(sqrt (+ (exp x) 1))
6.0ms
(+ (exp x) 1)

series68.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
54.0ms
(sqrt (+ (exp x) 1))
13.0ms
(+ (exp x) 1)

simplify368.0ms

Counts
12 → 30
Calls
12 calls:
Slowest
213.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
57.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
46.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
18.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
6.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune405.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

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

rewrite149.0ms

Algorithm
rewrite-expression-head
Rules
122×sqrt-div
61×flip-+
61×flip3-+
60×frac-times
30×associate-*l/
26×cbrt-div
23×pow1
18×associate-*r/
12×sqrt-prod
12×associate-*r*
12×add-sqr-sqrt
11×add-cube-cbrt
11×add-exp-log
11×pow1/2
11×add-cbrt-cube
11×*-un-lft-identity
9×pow-prod-up
7×pow-prod-down
7×associate-*l*
6×pow-plus
4×add-log-exp
4×cbrt-unprod
4×prod-exp
3×sqrt-unprod
3×pow2
2×*-commutative
2×pow3
1×rem-cbrt-cube
1×pow1/3
1×rem-square-sqrt
1×rem-sqrt-square
1×cbrt-prod
Counts
4 → 145
Calls
4 calls:
Slowest
61.0ms
(* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1)))
49.0ms
(cbrt (* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1))))
25.0ms
(* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1)))
7.0ms
(sqrt (+ (exp x) 1))

series163.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
69.0ms
(* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1)))
41.0ms
(cbrt (* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1))))
39.0ms
(sqrt (+ (exp x) 1))
14.0ms
(* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1)))

simplify13.9s

Counts
149 → 157
Calls
149 calls:
Slowest
810.0ms
(* (* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1))) (* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1))))
619.0ms
(* (* (+ (exp x) 1) (+ (exp x) 1)) (+ (exp x) 1))
618.0ms
(* (* (+ (exp x) 1) (+ (exp x) 1)) (+ (exp x) 1))
441.0ms
(* (* (* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1)))) (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1)))) (* (* (sqrt (+ (exp x) 1)) (sqrt (+ (exp x) 1))) (sqrt (+ (exp x) 1))))
392.0ms
(* (* (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (- (exp x) 1)))

prune2.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes47.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample17.1s

Algorithm
intervals
Results
6688×10.2s(body exit 10240)
12541×3.1s(body nan 80)
7383×2.1s(body real 80)
27354×680.0ms(pre true 80)
742×388.0ms(body real 160)