Average Error: 60.0 → 0.3
Time: 24.2s
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

Original60.0
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 60.0

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

  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 2019002 
(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: 22.5s)Debug log

sample493.0ms

Algorithm
intervals

simplify5.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

5.0ms
(- (/ 1 x) (/ 1 (tan x)))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.2b

localize37.0ms

Local error

Found 2 expressions with local error:

2.9b
(- (/ 1 x) (/ 1 (tan x)))
0.0b
(/ 1 (tan x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

6.0ms
(- (/ 1 x) (/ 1 (tan x)))
1.0ms
(/ 1 (tan x))

series135.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

86.0ms
(- (/ 1 x) (/ 1 (tan x)))
49.0ms
(/ 1 (tan x))

simplify264.0ms

Counts
20 → 41
Calls

20 calls. Slowest were:

142.0ms
(- (/ 1 x) (+ (* 1/3 x) (* 1/45 (pow x 3))))
30.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
25.0ms
(/ (exp (/ 1 x)) (exp (/ 1 (tan x))))

prune353.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize36.0ms

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)))

rewrite7.0ms

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))

series140.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

97.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
20.0ms
(* 2/945 (pow x 5))
15.0ms
(* 1/45 (pow x 3))
7.0ms
(* 1/3 x)

simplify159.0ms

Counts
20 → 55
Calls

20 calls. Slowest were:

31.0ms
(* (exp (* 1/45 (pow x 3))) (exp (* 2/945 (pow x 5))))
21.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
21.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))

prune544.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.3b

localize14.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* 1/3 x))
0.6b
(cbrt (* 1/3 x))
0.6b
(cbrt (* 1/3 x))
0.5b
(* (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)))
1.0ms
(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)) (cbrt (* 1/3 x)))
338.0ms
(cbrt (* 1/3 x))
324.0ms
(cbrt (* 1/3 x))
256.0ms
(cbrt (* 1/3 x))

simplify159.0ms

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))
16.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))

prune741.0ms

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.2b

localize16.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt x)
0.6b
(cbrt (* 1/3 x))
0.6b
(cbrt (* 1/3 x))
0.5b
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

11.0ms
(* (cbrt 1/3) (* (cbrt x) (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:

416.0ms
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))
299.0ms
(cbrt x)
293.0ms
(cbrt (* 1/3 x))
280.0ms
(cbrt (* 1/3 x))

simplify682.0ms

Counts
24 → 57
Calls

24 calls. Slowest were:

442.0ms
(* 1/3 (* (* (* (cbrt x) (cbrt (* 1/3 x))) (* (cbrt x) (cbrt (* 1/3 x)))) (* (cbrt x) (cbrt (* 1/3 x)))))
73.0ms
(* (* (cbrt -1/3) (* (cbrt 1/3) (cbrt -1))) (pow (pow x 2) 1/3))
22.0ms
(+ (log (cbrt 1/3)) (log (* (cbrt x) (cbrt (* 1/3 x)))))

prune651.0ms

Pruning

16 alts after pruning (15 fresh and 1 done)

Merged error: 0.1b

regimes122.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample15.4s

Algorithm
intervals