Average Error: 4.5 → 0.1
Time: 16.6s
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 4.5

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 15.9s)Debug log

sample493.0ms

Algorithm
intervals

simplify426.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
425.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune12.0ms

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)

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
5×add-sqr-sqrt
4×add-cube-cbrt
4×*-un-lft-identity
3×sqrt-prod
3×fma-def
2×add-log-exp
2×log1p-expm1-u
2×flip-+
2×add-exp-log
2×sqrt-div
2×add-cbrt-cube
2×flip3-+
2×pow1
2×expm1-log1p-u
1×pow1/2
1×rem-sqrt-square
1×+-commutative
Counts
2 → 31
Calls
2 calls:
Slowest
5.0ms
(sqrt (+ (exp x) 1))
3.0ms
(+ (exp x) 1)

series36.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
27.0ms
(sqrt (+ (exp x) 1))
9.0ms
(+ (exp x) 1)

simplify359.0ms

Counts
14 → 37
Calls
14 calls:
Slowest
185.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
72.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
54.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
16.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))
5.0ms
(sqrt (* (cbrt (+ (exp x) 1)) (cbrt (+ (exp x) 1))))

prune282.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize46.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.1b
(* (exp x) (exp x))
0.0b
(+ (pow (exp x) 3) (pow 1 3))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
14×*-un-lft-identity
11×add-cube-cbrt
11×fma-def
11×add-sqr-sqrt
10×pow1
8×add-log-exp
7×add-cbrt-cube
6×cube-prod
6×unpow-prod-down
4×log1p-expm1-u
4×add-exp-log
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
3×cube-mult
2×distribute-lft-out
2×unpow3
1×rem-cube-cbrt
1×flip--
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×sum-cubes
1×prod-exp
1×pow-exp
1×pow-prod-down
1×prod-diff
1×diff-log
1×fma-neg
1×distribute-rgt-out--
1×flip3--
1×pow-plus
1×flip3-+
1×pow-pow
1×sub-neg
1×sum-log
1×+-commutative
1×pow-to-exp
1×pow2
Counts
4 → 87
Calls
4 calls:
Slowest
7.0ms
(* (exp x) (exp x))
6.0ms
(+ (pow (exp x) 3) (pow 1 3))
2.0ms
(- (* 1 1) (* (exp x) 1))
1.0ms
(pow (exp x) 3)

series85.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
28.0ms
(+ (pow (exp x) 3) (pow 1 3))
24.0ms
(pow (exp x) 3)
16.0ms
(- (* 1 1) (* (exp x) 1))
16.0ms
(* (exp x) (exp x))

simplify920.0ms

Counts
51 → 99
Calls
51 calls:
Slowest
331.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
106.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))
87.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
54.0ms
(+ (* 3 x) (+ (* 9/2 (pow x 2)) 2))
40.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x)))

prune1.3s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.1b
(* (sqrt (exp x)) (exp x))
0.0b
(+ (pow (exp x) 3) (pow 1 3))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
15×*-un-lft-identity
13×add-sqr-sqrt
12×add-cube-cbrt
11×fma-def
8×add-log-exp
8×pow1
7×add-cbrt-cube
6×associate-*l*
6×cube-prod
6×unpow-prod-down
5×add-exp-log
4×log1p-expm1-u
4×expm1-log1p-u
3×sqrt-prod
3×associate-*r*
3×cube-mult
2×distribute-lft-out
2×unpow3
2×pow1/2
1×rem-cube-cbrt
1×flip--
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×sum-cubes
1×prod-exp
1×pow-exp
1×pow-prod-down
1×prod-diff
1×cube-unmult
1×diff-log
1×fma-neg
1×distribute-rgt-out--
1×flip3--
1×pow-plus
1×flip3-+
1×pow-pow
1×sub-neg
1×sum-log
1×+-commutative
1×pow-to-exp
Counts
4 → 90
Calls
4 calls:
Slowest
12.0ms
(+ (pow (exp x) 3) (pow 1 3))
8.0ms
(* (sqrt (exp x)) (exp x))
2.0ms
(- (* 1 1) (* (exp x) 1))
1.0ms
(pow (exp x) 3)

series90.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
30.0ms
(- (* 1 1) (* (exp x) 1))
27.0ms
(* (sqrt (exp x)) (exp x))
17.0ms
(pow (exp x) 3)
16.0ms
(+ (pow (exp x) 3) (pow 1 3))

simplify830.0ms

Counts
54 → 102
Calls
54 calls:
Slowest
262.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
82.0ms
(* (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))) (* (* (exp x) (exp x)) (exp x)))
77.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
48.0ms
(+ (* 3/2 x) (+ (* 9/8 (pow x 2)) 1))
35.0ms
(+ (* 3 x) (+ (* 9/2 (pow x 2)) 1))

prune1.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes39.0ms

Accuracy

0% (0.0b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch1.0ms

end0.0ms

sample9.3s

Algorithm
intervals