Average Error: 4.1 → 0.0
Time: 14.8s
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 4.1

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
  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-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))

Details

Time bar (total: 14.3s)Debug log

sample542.0ms

Algorithm
intervals

simplify87.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

87.0ms
(sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 2 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

4.0ms
(+ (exp x) 1)
4.0ms
(sqrt (+ (exp x) 1))

series56.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

31.0ms
(sqrt (+ (exp x) 1))
24.0ms
(+ (exp x) 1)

simplify351.0ms

Counts
14 → 37
Calls

14 calls. Slowest were:

191.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
59.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
43.0ms
(sqrt (- (* (exp x) (exp x)) (* 1 1)))

prune350.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

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))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

1.0ms
(sqrt (fma (sqrt (exp x)) (sqrt (exp x)) 1))
1.0ms
(sqrt (exp x))
1.0ms
(sqrt (exp x))

series133.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

63.0ms
(sqrt (fma (sqrt (exp x)) (sqrt (exp x)) 1))
44.0ms
(fma (sqrt (exp x)) (sqrt (exp x)) 1)
14.0ms
(sqrt (exp x))
12.0ms
(sqrt (exp x))

simplify339.0ms

Counts
24 → 64
Calls

24 calls. Slowest were:

155.0ms
(- (+ (sqrt 2) (+ (* 1/4 (/ (pow x 2) (sqrt 2))) (* 1/2 (/ x (sqrt 2))))) (* 1/8 (/ (pow x 2) (pow (sqrt 2) 3))))
53.0ms
(+ (* 1/2 x) (+ (* 1/8 (pow x 2)) 1))
43.0ms
(+ (* 1/2 x) (+ (* 1/8 (pow x 2)) 1))

prune697.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

4.5b
(- (* 1 1) (* (exp x) 1))
0.1b
(pow (exp x) 3)
0.0b
(* (exp x) (exp x))
0.0b
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

7.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
4.0ms
(* (exp x) (exp x))
2.0ms
(- (* 1 1) (* (exp x) 1))

series59.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

19.0ms
(+ (* (exp x) (exp x)) (- (* 1 1) (* (exp x) 1)))
14.0ms
(pow (exp x) 3)
14.0ms
(* (exp x) (exp x))
12.0ms
(- (* 1 1) (* (exp x) 1))

simplify1.0s

Counts
43 → 91
Calls

43 calls. Slowest were:

261.0ms
(- (+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))))
167.0ms
(* (exp (* (exp x) (exp x))) (exp (- (* 1 1) (* (exp x) 1))))
131.0ms
(* (exp (* (exp x) (exp x))) (/ (exp (* 1 1)) (exp (* (exp x) 1))))

prune1.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes45.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.4s

Algorithm
intervals