Average Error: 59.9 → 0.3
Time: 22.8s
Precision: 64
Internal Precision: 128
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{1}{3} \cdot x + \left({x}^{5} \cdot \frac{2}{945} + \frac{1}{45} \cdot {x}^{3}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

    \[\frac{1}{x} - \frac{1}{\tan x}\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

  3. Final simplification0.3

    \[\leadsto \frac{1}{3} \cdot x + \left({x}^{5} \cdot \frac{2}{945} + \frac{1}{45} \cdot {x}^{3}\right)\]

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))

Details

Time bar (total: 21.3s)Debug log

sample562.0ms

Algorithm
intervals

simplify9.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

9.0ms
(- (/ 1 x) (/ 1 (tan x)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.7b

localize22.0ms

Local error

Found 2 expressions with local error:

3.1b
(- (/ 1 x) (/ 1 (tan x)))
0.0b
(/ 1 (tan x))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

7.0ms
(- (/ 1 x) (/ 1 (tan x)))
2.0ms
(/ 1 (tan x))

series113.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

88.0ms
(- (/ 1 x) (/ 1 (tan x)))
25.0ms
(/ 1 (tan x))

simplify265.0ms

Counts
20 → 41
Calls

20 calls. Slowest were:

138.0ms
(- (/ 1 x) (+ (* 1/3 x) (* 1/45 (pow x 3))))
53.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
13.0ms
(/ (exp (/ 1 x)) (exp (/ 1 (tan x))))

prune288.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize32.0ms

Local error

Found 4 expressions with local error:

0.3b
(* 1/3 x)
0.1b
(* 1/45 (pow x 3))
0.1b
(* 2/945 (pow x 5))
0.0b
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 43
Calls

4 calls. Slowest were:

3.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
1.0ms
(* 1/45 (pow x 3))
1.0ms
(* 2/945 (pow x 5))

series139.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
24.0ms
(* 2/945 (pow x 5))
23.0ms
(* 1/45 (pow x 3))
7.0ms
(* 1/3 x)

simplify175.0ms

Counts
20 → 55
Calls

20 calls. Slowest were:

34.0ms
(* (exp (* 1/45 (pow x 3))) (exp (* 2/945 (pow x 5))))
25.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
24.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))

prune574.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.2b

localize12.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt (* 1/3 x))
0.7b
(cbrt (* 1/3 x))
0.7b
(cbrt (* 1/3 x))
0.4b
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

4.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
0.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt (* 1/3 x))

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

378.0ms
(cbrt (* 1/3 x))
359.0ms
(cbrt (* 1/3 x))
357.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
345.0ms
(cbrt (* 1/3 x))

simplify130.0ms

Counts
31 → 63
Calls

31 calls. Slowest were:

18.0ms
(* (pow (cbrt -1/3) 2) (pow (pow x 2) 1/3))
17.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
17.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))

prune806.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.2b

localize11.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt (* 1/3 x))
0.7b
(cbrt (* 1/3 x))
0.7b
(cbrt (* 1/3 x))
0.7b
(cbrt (* 1/3 x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 36
Calls

4 calls. Slowest were:

1.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt (* 1/3 x))

series1.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

357.0ms
(cbrt (* 1/3 x))
332.0ms
(cbrt (* 1/3 x))
296.0ms
(cbrt (* 1/3 x))
271.0ms
(cbrt (* 1/3 x))

simplify67.0ms

Counts
16 → 48
Calls

16 calls. Slowest were:

10.0ms
(* (cbrt -1/3) (pow (* x -1) 1/3))
10.0ms
(* (cbrt -1/3) (pow (* x -1) 1/3))
7.0ms
(* (cbrt -1/3) (pow (* x -1) 1/3))

prune696.0ms

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.1b

regimes121.0ms

Accuracy

0% (0.1b remaining)

Error of 0.3b against oracle of 0.2b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample14.5s

Algorithm
intervals