Average Error: 39.5 → 0.0
Time: 16.5s
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 39.5

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

  2. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 6.0s)Debug log

start581.0ms

Algorithm
intervals

setup250.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 2 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

11.0ms
(sqrt (+ (exp x) 1))
6.0ms
(+ (exp x) 1)

series71.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

60.0ms
(sqrt (+ (exp x) 1))
10.0ms
(+ (exp x) 1)

simplify373.0ms

Counts
14 → 37
Calls

14 calls. Slowest were:

225.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
70.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
31.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune310.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

14.0ms
(* (cbrt (sqrt (+ (exp x) 1))) (cbrt (sqrt (+ (exp x) 1))))
4.0ms
(cbrt (sqrt (+ (exp x) 1)))
4.0ms
(cbrt (sqrt (+ (exp x) 1)))

series602.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

168.0ms
(cbrt (sqrt (+ (exp x) 1)))
163.0ms
(cbrt (sqrt (+ (exp x) 1)))
141.0ms
(cbrt (sqrt (+ (exp x) 1)))
129.0ms
(* (cbrt (sqrt (+ (exp x) 1))) (cbrt (sqrt (+ (exp x) 1))))

simplify2.4s

Counts
74 → 110
Calls

74 calls. Slowest were:

431.0ms
(+ (* 1/12 (* x (pow 2 1/6))) (+ (pow 2 1/6) (* 7/288 (* (pow x 2) (pow 2 1/6)))))
408.0ms
(+ (* 1/12 (* x (pow 2 1/6))) (+ (pow 2 1/6) (* 7/288 (* (pow x 2) (pow 2 1/6)))))
349.0ms
(+ (* 1/12 (* x (pow 2 1/6))) (+ (pow 2 1/6) (* 7/288 (* (pow x 2) (pow 2 1/6)))))

prune1.3s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes21.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch1.0ms