Average Error: 4.2 → 0.1
Time: 14.5s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{(\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + 1)_*}\]

Error

Bits error versus x

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 add-sqr-sqrt0.1

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

  5. Applied fma-def0.1

    \[\leadsto \sqrt{\color{blue}{(\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + 1)_*}}\]

  6. Final simplification0.1

    \[\leadsto \sqrt{(\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + 1)_*}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 14.0s)Debug log

sample413.0ms

Algorithm
intervals

simplify91.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

90.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune10.0ms

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

8.0ms
(+ (exp x) 1)
6.0ms
(sqrt (+ (exp x) 1))

series48.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

29.0ms
(+ (exp x) 1)
19.0ms
(sqrt (+ (exp x) 1))

simplify278.0ms

Counts
14 → 37
Calls

14 calls. Slowest were:

164.0ms
(- (+ (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)))
33.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune348.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

2.0ms
(sqrt (exp x))
2.0ms
(sqrt (fma (sqrt (exp x)) (sqrt (exp x)) 1))
2.0ms
(sqrt (exp x))

series158.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(sqrt (fma (sqrt (exp x)) (sqrt (exp x)) 1))
50.0ms
(fma (sqrt (exp x)) (sqrt (exp x)) 1)
20.0ms
(sqrt (exp x))
20.0ms
(sqrt (exp x))

simplify311.0ms

Counts
24 → 64
Calls

24 calls. Slowest were:

156.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
43.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
33.0ms
(+ (* 1/2 x) (+ (* 1/8 (pow x 2)) 1))

prune560.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))
0.0b
(* (fabs (cbrt (exp x))) (sqrt (cbrt (exp x))))
0.0b
(sqrt (exp x))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

10.0ms
(* (fabs (cbrt (exp x))) (sqrt (cbrt (exp x))))
2.0ms
(sqrt (exp x))
1.0ms
(cbrt (exp x))

series450.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(* (fabs (cbrt (exp x))) (sqrt (cbrt (exp x))))
145.0ms
(cbrt (exp x))
132.0ms
(cbrt (exp x))
20.0ms
(sqrt (exp x))

simplify697.0ms

Counts
36 → 77
Calls

36 calls. Slowest were:

345.0ms
(+ (* 1/72 (* (pow x 2) (fabs (pow (exp x) 1/3)))) (+ (fabs (pow (exp x) 1/3)) (* 1/6 (* x (fabs (pow (exp x) 1/3))))))
87.0ms
(* (* (* (fabs (cbrt (exp x))) (fabs (cbrt (exp x)))) (fabs (cbrt (exp x)))) (* (* (sqrt (cbrt (exp x))) (sqrt (cbrt (exp x)))) (sqrt (cbrt (exp x)))))
70.0ms
(+ (* 1/3 x) (+ (* 1/18 (pow x 2)) 1))

prune911.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes40.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample9.6s

Algorithm
intervals