Average Error: 4.2 → 0.2
Time: 2.7m
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.2

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  3. Using strategy rm

  4. Applied flip3-+0.2

    \[\leadsto \sqrt{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}\]

  5. Final simplification0.2

    \[\leadsto \sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 2.7m)Debug log

sample752.0ms

Algorithm
intervals

simplify147.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

146.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune23.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize66.0ms

Local error

Found 2 expressions with local error:

0.3b
(sqrt (+ (exp x) 1))
0.0b
(+ (exp x) 1)

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

12.0ms
(sqrt (+ (exp x) 1))
8.0ms
(+ (exp x) 1)

series109.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

95.0ms
(sqrt (+ (exp x) 1))
14.0ms
(+ (exp x) 1)

simplify392.0ms

Counts
14 → 37
Calls

14 calls. Slowest were:

188.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
81.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))
56.0ms
(sqrt (+ (pow (exp x) 3) (pow 1 3)))

prune246.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.2b

localize38.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (* 1 1) (* (exp x) 1))
0.3b
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (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))))
0.2b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

rewrite57.0ms

Algorithm
rewrite-expression-head
Counts
4 → 118
Calls

4 calls. Slowest were:

28.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
16.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
8.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

series247.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

123.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
78.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))
26.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
20.0ms
(- (* 1 1) (* (exp x) 1))

simplify9.9s

Counts
98 → 130
Calls

98 calls. Slowest were:

744.0ms
(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))))))
407.0ms
(sqrt (/ (+ (pow (exp x) 3) (pow 1 3)) (+ (pow (* (exp x) (exp x)) 3) (pow (- (* 1 1) (* (exp x) 1)) 3))))
377.0ms
(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)))))))

prune1.7s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize1.0ms

Local error

Found 2 expressions with local error:

0.3b
(sqrt (+ (exp x) 1))
0.0b
(+ (exp x) 1)

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

6.0ms
(sqrt (+ (exp x) 1))
5.0ms
(+ (exp x) 1)

series35.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

28.0ms
(sqrt (+ (exp x) 1))
7.0ms
(+ (exp x) 1)

simplify247.0ms

Counts
14 → 37
Calls

14 calls. Slowest were:

140.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
37.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
35.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))

prune306.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

localize23.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (* 1 1) (* (exp x) 1))
0.3b
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (fma (exp x) (expm1 x) 1))))
0.2b
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (fma (exp x) (expm1 x) 1)))
0.2b
(/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))

rewrite130.0ms

Algorithm
rewrite-expression-head
Counts
4 → 674
Calls

4 calls. Slowest were:

42.0ms
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (fma (exp x) (expm1 x) 1)))
35.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
28.0ms
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (fma (exp x) (expm1 x) 1))))

series557.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

282.0ms
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (fma (exp x) (expm1 x) 1))))
153.0ms
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1))))) (sqrt (fma (exp x) (expm1 x) 1)))
104.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))))
19.0ms
(- (* 1 1) (* (exp x) 1))

simplify2.1m

Counts
923 → 686
Calls

923 calls. Slowest were:

876.0ms
(sqrt (/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1)))))) (* (cbrt (sqrt (fma (exp x) (expm1 x) 1))) (cbrt (sqrt (fma (exp x) (expm1 x) 1))))))
808.0ms
(/ (/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1)))))) (* (cbrt (sqrt (fma (exp x) (expm1 x) 1))) (cbrt (sqrt (fma (exp x) (expm1 x) 1)))))
806.0ms
(/ (+ (pow (exp x) 3) (pow 1 3)) (sqrt (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (- (* 1 1) (* (exp x) 1)) (- (* 1 1) (* (exp x) 1))))))

prune12.4s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.2b

regimes43.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 0.2b

bsearch2.0ms

end0.0ms

sample9.6s

Algorithm
intervals