Average Error: 52.5 → 0.2
Time: 18.3s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0817179295038961:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9597911148379592:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{3}{40}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + x \cdot 2\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.5
Target44.8
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0817179295038961

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)}\]

    if -1.0817179295038961 < x < 0.9597911148379592

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    if 0.9597911148379592 < x

    1. Initial program 31.1

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0817179295038961:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9597911148379592:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{3}{40}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + x \cdot 2\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))

Details

Time bar (total: 17.8s)Debug log

sample128.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 52.7b

localize23.0ms

Local error

Found 3 expressions with local error:

13.8b
(sqrt (+ (* x x) 1))
3.3b
(log (+ x (sqrt (+ (* x x) 1))))
0.5b
(+ x (sqrt (+ (* x x) 1)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

3.0ms
(sqrt (+ (* x x) 1))
2.0ms
(log (+ x (sqrt (+ (* x x) 1))))
2.0ms
(+ x (sqrt (+ (* x x) 1)))

series185.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

130.0ms
(log (+ x (sqrt (+ (* x x) 1))))
38.0ms
(+ x (sqrt (+ (* x x) 1)))
17.0ms
(sqrt (+ (* x x) 1))

simplify1.5s

Counts
21 → 47
Calls

21 calls. Slowest were:

350.0ms
(- (+ (log (/ -1 x)) (+ (log 1/2) (* 3/32 (/ 1 (pow x 4))))) (* 1/4 (/ 1 (pow x 2))))
231.0ms
(- (+ (log 2) (* 1/4 (/ 1 (pow x 2)))) (+ (log (/ 1 x)) (* 3/32 (/ 1 (pow x 4)))))
153.0ms
(- (* 1/8 (/ 1 (pow x 3))) (+ (* 1/16 (/ 1 (pow x 5))) (* 1/2 (/ 1 x))))

prune410.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize20.0ms

Local error

Found 3 expressions with local error:

18.2b
(- (+ x (* 3/40 (pow x 5))) (* 1/6 (pow x 3)))
0.3b
(* 1/6 (pow x 3))
0.0b
(* 3/40 (pow x 5))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

9.0ms
(- (+ x (* 3/40 (pow x 5))) (* 1/6 (pow x 3)))
2.0ms
(* 1/6 (pow x 3))
2.0ms
(* 3/40 (pow x 5))

series80.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

28.0ms
(* 3/40 (pow x 5))
27.0ms
(- (+ x (* 3/40 (pow x 5))) (* 1/6 (pow x 3)))
24.0ms
(* 1/6 (pow x 3))

simplify420.0ms

Counts
17 → 46
Calls

17 calls. Slowest were:

148.0ms
(/ (exp (+ x (* 3/40 (pow x 5)))) (exp (* 1/6 (pow x 3))))
85.0ms
(- (+ x (* 3/40 (pow x 5))) (* 1/6 (pow x 3)))
73.0ms
(- (+ x (* 3/40 (pow x 5))) (* 1/6 (pow x 3)))

prune384.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

19.8b
(- (/ -1/16 (pow x 5)) (- (/ 1/2 x) (/ (/ 1/8 x) (* x x))))
12.3b
(log (- (/ -1/16 (pow x 5)) (- (/ 1/2 x) (/ (/ 1/8 x) (* x x)))))
0.2b
(/ -1/16 (pow x 5))
0.1b
(/ (/ 1/8 x) (* x x))

rewrite54.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

28.0ms
(- (/ -1/16 (pow x 5)) (- (/ 1/2 x) (/ (/ 1/8 x) (* x x))))
18.0ms
(log (- (/ -1/16 (pow x 5)) (- (/ 1/2 x) (/ (/ 1/8 x) (* x x)))))
5.0ms
(/ (/ 1/8 x) (* x x))

series154.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

83.0ms
(log (- (/ -1/16 (pow x 5)) (- (/ 1/2 x) (/ (/ 1/8 x) (* x x)))))
30.0ms
(- (/ -1/16 (pow x 5)) (- (/ 1/2 x) (/ (/ 1/8 x) (* x x))))
22.0ms
(/ (/ 1/8 x) (* x x))
19.0ms
(/ -1/16 (pow x 5))

simplify4.6s

Counts
54 → 89
Calls

54 calls. Slowest were:

544.0ms
(/ (* (* (/ 1/8 x) (/ 1/8 x)) (/ 1/8 x)) (* (* (* x x) (* x x)) (* x x)))
394.0ms
(- (+ (log (/ -1 x)) (+ (log 1/2) (* 3/32 (/ 1 (pow x 4))))) (* 1/4 (/ 1 (pow x 2))))
377.0ms
(- (+ (log (/ 1 x)) (+ (* 3/32 (/ 1 (pow x 4))) (log -1/2))) (* 1/4 (/ 1 (pow x 2))))

prune798.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize22.0ms

Local error

Found 3 expressions with local error:

9.4b
(log (- (+ (* 2 x) (/ 1/2 x)) (/ (/ 1/8 x) (* x x))))
0.1b
(/ (/ 1/8 x) (* x x))
0.0b
(+ (* 2 x) (/ 1/2 x))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
3 → 52
Calls

3 calls. Slowest were:

16.0ms
(log (- (+ (* 2 x) (/ 1/2 x)) (/ (/ 1/8 x) (* x x))))
4.0ms
(/ (/ 1/8 x) (* x x))
3.0ms
(+ (* 2 x) (/ 1/2 x))

series140.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(log (- (+ (* 2 x) (/ 1/2 x)) (/ (/ 1/8 x) (* x x))))
38.0ms
(+ (* 2 x) (/ 1/2 x))
15.0ms
(/ (/ 1/8 x) (* x x))

simplify3.2s

Counts
33 → 61
Calls

33 calls. Slowest were:

675.0ms
(log (- (* (+ (pow (* 2 x) 3) (pow (/ 1/2 x) 3)) (* x x)) (* (+ (* (* 2 x) (* 2 x)) (- (* (/ 1/2 x) (/ 1/2 x)) (* (* 2 x) (/ 1/2 x)))) (/ 1/8 x))))
543.0ms
(/ (* (* (/ 1/8 x) (/ 1/8 x)) (/ 1/8 x)) (* (* (* x x) (* x x)) (* x x)))
375.0ms
(log (- (pow (+ (* 2 x) (/ 1/2 x)) 3) (pow (/ (/ 1/8 x) (* x x)) 3)))

prune525.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes45.0ms

Accuracy

100% (0.0b remaining)

Error of 0.2b against oracle of 0.2b and baseline of 31.5b

bsearch134.0ms

end0.0ms

sample4.9s

Algorithm
intervals