Average Error: 31.7 → 0.2
Time: 39.0s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) + \frac{\frac{\frac{-1}{8}}{x}}{x \cdot x}\right)\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  2. Simplified31.7

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

  3. Taylor expanded around inf 0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  4. Simplified0.2

    \[\leadsto \log \left(x + \color{blue}{\left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \left(x + \frac{\frac{-1}{2}}{x}\right)\right)}\right)\]

  5. Final simplification0.2

    \[\leadsto \log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) + \frac{\frac{\frac{-1}{8}}{x}}{x \cdot x}\right)\right)\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Details

Time bar (total: 38.5s)Debug log

sample146.0ms

Algorithm
intervals

simplify109.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

109.0ms
(log (+ x (sqrt (- (* x x) 1))))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 34.1b

localize12.0ms

Local error

Found 2 expressions with local error:

32.7b
(sqrt (fma x x -1))
0.2b
(log (+ x (sqrt (fma x x -1))))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

1.0ms
(log (+ x (sqrt (fma x x -1))))
1.0ms
(sqrt (fma x x -1))

series122.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

99.0ms
(log (+ x (sqrt (fma x x -1))))
23.0ms
(sqrt (fma x x -1))

simplify2.3s

Counts
16 → 36
Calls

16 calls. Slowest were:

395.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
362.0ms
(- (log 2) (+ (log (/ 1 x)) (+ (* 3/32 (/ 1 (pow x 4))) (* 1/4 (/ 1 (pow x 2))))))
351.0ms
(+ (log -1/2) (+ (log (/ -1 x)) (+ (* 3/32 (/ 1 (pow x 4))) (* 1/4 (/ 1 (pow x 2))))))

prune294.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ -1/4 (* x x))
0.2b
(/ 3/32 (pow x 4))
0.2b
(+ (log x) (log 2))
0.0b
(- (/ -1/4 (* x x)) (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

40.0ms
(- (/ -1/4 (* x x)) (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))
3.0ms
(+ (log x) (log 2))
0.0ms
(/ 3/32 (pow x 4))

series507.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

361.0ms
(+ (log x) (log 2))
107.0ms
(- (/ -1/4 (* x x)) (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))
26.0ms
(/ 3/32 (pow x 4))
13.0ms
(/ -1/4 (* x x))

simplify21.9s

Counts
104 → 128
Calls

104 calls. Slowest were:

619.0ms
(- (+ (log 2) (log x)) (+ (* 1/4 (/ 1 (pow x 2))) (* 3/32 (/ 1 (pow x 4)))))
542.0ms
(fma (- (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))) (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2)))) (* (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2)))) (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))))
525.0ms
(fma (- (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))) (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2)))) (* (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2)))) (sqrt (- (/ 3/32 (pow x 4)) (+ (log x) (log 2))))))

prune1.6s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize36.0ms

Local error

Found 3 expressions with local error:

0.2b
(log (+ x (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x)))))
0.1b
(/ (/ -1/8 x) (* x x))
0.0b
(+ x (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

14.0ms
(+ x (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x))))
9.0ms
(log (+ x (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x)))))
6.0ms
(/ (/ -1/8 x) (* x x))

series154.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

100.0ms
(log (+ x (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x)))))
30.0ms
(+ x (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x))))
24.0ms
(/ (/ -1/8 x) (* x x))

simplify3.3s

Counts
28 → 62
Calls

28 calls. Slowest were:

842.0ms
(log (+ (pow x 3) (pow (+ (/ (/ -1/8 x) (* x x)) (+ x (/ -1/2 x))) 3)))
583.0ms
(/ (* (* (/ -1/8 x) (/ -1/8 x)) (/ -1/8 x)) (* (* (* x x) (* x x)) (* x x)))
329.0ms
(- (+ (log -1/8) (* 4 (pow x 2))) (+ (* 24 (pow x 4)) (* 3 (log x))))

prune496.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize43.0ms

Local error

Found 4 expressions with local error:

1.0b
(pow (log 2) 3)
1.0b
(* (log 2) (log 2))
0.6b
(pow (log x) 3)
0.5b
(* (log x) (log x))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

4.0ms
(* (log x) (log x))
4.0ms
(* (log 2) (log 2))
1.0ms
(pow (log 2) 3)

series374.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

188.0ms
(pow (log x) 3)
185.0ms
(* (log x) (log x))
0.0ms
(pow (log 2) 3)
0.0ms
(* (log 2) (log 2))

simplify1.3s

Counts
46 → 94
Calls

46 calls. Slowest were:

516.0ms
(pow (- (log -1) (log (/ -1 x))) 2)
451.0ms
(pow (- (log -1) (log (/ -1 x))) 3)
137.0ms
(* -1 (pow (log (/ 1 x)) 3))

prune1.6s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes46.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch2.0ms

end0.0ms

sample4.2s

Algorithm
intervals