Average Error: 31.4 → 0.1
Time: 25.1s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \left(\sqrt{\sqrt{x - 1}} \cdot \sqrt{\sqrt{x - 1}}\right)\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied difference-of-sqr-131.4

    \[\leadsto \log \left(x + \sqrt{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}}\right)\]

  4. Applied sqrt-prod0.1

    \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x - 1}}\right)\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.1

    \[\leadsto \log \left(x + \sqrt{x + 1} \cdot \sqrt{\color{blue}{\sqrt{x - 1} \cdot \sqrt{x - 1}}}\right)\]

  7. Applied sqrt-prod0.1

    \[\leadsto \log \left(x + \sqrt{x + 1} \cdot \color{blue}{\left(\sqrt{\sqrt{x - 1}} \cdot \sqrt{\sqrt{x - 1}}\right)}\right)\]

  8. Final simplification0.1

    \[\leadsto \log \left(x + \sqrt{1 + x} \cdot \left(\sqrt{\sqrt{x - 1}} \cdot \sqrt{\sqrt{x - 1}}\right)\right)\]

Reproduce

herbie shell --seed 2018365 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Details

Time bar (total: 19.5s)Debug log

start157.0ms

Algorithm
intervals

setup138.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 33.2b

localize26.0ms

Local error

Found 3 expressions with local error:

31.9b
(sqrt (- (* x x) 1))
0.2b
(log (+ x (sqrt (- (* x x) 1))))
0.0b
(+ x (sqrt (- (* x x) 1)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 39
Calls

3 calls. Slowest were:

3.0ms
(sqrt (- (* x x) 1))
3.0ms
(log (+ x (sqrt (- (* x x) 1))))
3.0ms
(+ x (sqrt (- (* x x) 1)))

series150.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(log (+ x (sqrt (- (* x x) 1))))
37.0ms
(sqrt (- (* x x) 1))
26.0ms
(+ x (sqrt (- (* x x) 1)))

simplify2.7s

Counts
22 → 48
Calls

22 calls. Slowest were:

383.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
273.0ms
(- (+ (log (sqrt -1)) (/ x (sqrt -1))) (* 1/6 (/ (pow x 3) (pow (sqrt -1) 3))))
271.0ms
(- (log 2) (+ (log (/ 1 x)) (+ (* 3/32 (/ 1 (pow x 4))) (* 1/4 (/ 1 (pow x 2))))))

prune519.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (+ x 1)) (sqrt (- x 1)))
0.2b
(log (+ x (* (sqrt (+ x 1)) (sqrt (- x 1)))))
0.0b
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))
0.0b
(sqrt (- x 1))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
1.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))
1.0ms
(log (+ x (* (sqrt (+ x 1)) (sqrt (- x 1)))))

series182.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(log (+ x (* (sqrt (+ x 1)) (sqrt (- x 1)))))
41.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
32.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))
14.0ms
(sqrt (- x 1))

simplify3.5s

Counts
52 → 83
Calls

52 calls. Slowest were:

550.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))))
534.0ms
(log (- (* x x) (* (* (sqrt (+ x 1)) (sqrt (- x 1))) (* (sqrt (+ x 1)) (sqrt (- x 1))))))
466.0ms
(- (+ (* 1/2 (/ x (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 2) (pow (sqrt -1) 3))))

prune568.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (+ x 1)) (* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1)))))
0.5b
(* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1))))
0.2b
(log (+ x (* (sqrt (+ x 1)) (* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1)))))))
0.1b
(sqrt (sqrt (- x 1)))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

20.0ms
(* (sqrt (+ x 1)) (* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1)))))
6.0ms
(* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1))))
1.0ms
(sqrt (sqrt (- x 1)))

series144.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

62.0ms
(log (+ x (* (sqrt (+ x 1)) (* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1)))))))
38.0ms
(sqrt (sqrt (- x 1)))
30.0ms
(* (sqrt (+ x 1)) (* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1)))))
13.0ms
(* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1))))

simplify5.2s

Counts
124 → 137
Calls

124 calls. Slowest were:

518.0ms
(log (+ (pow x 3) (pow (* (sqrt (+ x 1)) (* (sqrt (sqrt (- x 1))) (sqrt (sqrt (- x 1))))) 3)))
423.0ms
(- (+ (* 1/2 (/ x (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 2) (pow (sqrt -1) 3))))
395.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))

prune1.9s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

1.2b
(/ -1/4 (* x x))
0.5b
(/ -3/32 (pow x 4))
0.2b
(+ (log 2) (log x))
0.0b
(+ (+ (log 2) (log x)) (/ -3/32 (pow x 4)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

17.0ms
(+ (+ (log 2) (log x)) (/ -3/32 (pow x 4)))
3.0ms
(+ (log 2) (log x))
0.0ms
(/ -3/32 (pow x 4))

series1.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.3s
(+ (+ (log 2) (log x)) (/ -3/32 (pow x 4)))
315.0ms
(+ (log 2) (log x))
24.0ms
(/ -3/32 (pow x 4))
18.0ms
(/ -1/4 (* x x))

simplify1.5s

Counts
26 → 65
Calls

26 calls. Slowest were:

396.0ms
(+ (* (- (* (log 2) (log 2)) (* (log x) (log x))) (pow x 4)) (* (- (log 2) (log x)) -3/32))
297.0ms
(- (+ (log 2) (log -1)) (+ (log (/ -1 x)) (* 3/32 (/ 1 (pow x 4)))))
187.0ms
(+ (* (+ (pow (log 2) 3) (pow (log x) 3)) (pow x 4)) (* (+ (* (log 2) (log 2)) (- (* (log x) (log x)) (* (log 2) (log x)))) -3/32))

prune928.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes38.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms