Average Error: 59.9 → 0.3
Time: 23.8s
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 2019022 
(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

sample540.0ms

Algorithm
intervals
Results
336.0ms121×body2560valid
100.0ms72×body1280valid
45.0ms256×pre80true
30.0ms38×body640valid
12.0ms20×body320valid
1.0ms5×body160valid

simplify9.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
9.0ms
(- (/ 1 x) (/ 1 (tan x)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.8b

localize21.0ms

Local error

Found 2 expressions with local error:

1.6b
(- (/ 1 x) (/ 1 (tan x)))
0.0b
(/ 1 (tan x))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
5×div-inv
5×add-sqr-sqrt
4×add-log-exp
4×distribute-lft-out--
3×add-cube-cbrt
3×add-exp-log
3×associate-/r*
3×pow1
2×add-cbrt-cube
1×difference-of-squares
1×inv-pow
1×flip--
1×pow-flip
1×frac-sub
1×associate-/r/
1×tan-quot
1×diff-log
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
1×rec-exp
Counts
2 → 35
Calls
2 calls:
Slowest
8.0ms
(- (/ 1 x) (/ 1 (tan x)))
2.0ms
(/ 1 (tan x))

series149.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
104.0ms
(- (/ 1 x) (/ 1 (tan x)))
45.0ms
(/ 1 (tan x))

simplify259.0ms

Counts
20 → 41
Calls
20 calls:
Slowest
121.0ms
(- (/ 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))))
19.0ms
(/ (exp (/ 1 x)) (exp (/ 1 (tan x))))
9.0ms
(- (/ 1 x) (/ (cos x) (sin x)))
9.0ms
(- (/ 1 x) (/ 1 (tan x)))

prune297.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize32.0ms

Local error

Found 4 expressions with local error:

0.4b
(* 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)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
6×add-log-exp
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×*-commutative
1×flip-+
1×unpow3
1×flip3-+
1×sum-log
1×+-commutative
1×cube-mult
Counts
4 → 43
Calls
4 calls:
Slowest
7.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
1.0ms
(* 1/45 (pow x 3))
1.0ms
(* 2/945 (pow x 5))
0.0ms
(* 1/3 x)

series127.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
73.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
27.0ms
(* 2/945 (pow x 5))
20.0ms
(* 1/45 (pow x 3))
7.0ms
(* 1/3 x)

simplify130.0ms

Counts
20 → 55
Calls
20 calls:
Slowest
32.0ms
(* (exp (* 1/45 (pow x 3))) (exp (* 2/945 (pow x 5))))
18.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
12.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
11.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
10.0ms
(* 1/45 (pow x 3))

prune564.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.3b

localize12.0ms

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
7×pow1/3
6×add-cube-cbrt
6×add-exp-log
6×*-un-lft-identity
6×add-sqr-sqrt
5×cbrt-prod
4×add-log-exp
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 51
Calls
4 calls:
Slowest
8.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
0.0ms
(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
348.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
321.0ms
(cbrt (* 1/3 x))
304.0ms
(cbrt (* 1/3 x))
289.0ms
(cbrt (* 1/3 x))

simplify146.0ms

Counts
31 → 63
Calls
31 calls:
Slowest
12.0ms
(* (cbrt -1/3) (pow (* x -1) 1/3))
10.0ms
(* (cbrt -1/3) (pow (* x -1) 1/3))
9.0ms
(* (* 1/3 x) (* 1/3 x))
9.0ms
(* (pow (cbrt -1/3) 2) (pow (pow x 2) 1/3))
9.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))

prune771.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.2b

localize39.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt x)
0.6b
(cbrt (* 1/3 x))
0.6b
(cbrt (* 1/3 x))
0.4b
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
9×pow1
6×pow1/3
5×add-cube-cbrt
5×pow-prod-down
5×add-cbrt-cube
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
3×cbrt-unprod
3×prod-exp
3×associate-*l*
2×cbrt-prod
1×*-commutative
1×associate-*r*
Counts
4 → 45
Calls
4 calls:
Slowest
18.0ms
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))
1.0ms
(cbrt (* 1/3 x))
1.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt x)

series1.3s

Counts
4 → 12
Calls
4 calls:
Slowest
412.0ms
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))
299.0ms
(cbrt (* 1/3 x))
294.0ms
(cbrt (* 1/3 x))
253.0ms
(cbrt x)

simplify565.0ms

Counts
24 → 57
Calls
24 calls:
Slowest
341.0ms
(* 1/3 (* (* (* (cbrt x) (cbrt (* 1/3 x))) (* (cbrt x) (cbrt (* 1/3 x)))) (* (cbrt x) (cbrt (* 1/3 x)))))
59.0ms
(* (* (cbrt -1/3) (* (cbrt 1/3) (cbrt -1))) (pow (pow x 2) 1/3))
21.0ms
(* 1/3 (* x (* 1/3 x)))
17.0ms
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))
17.0ms
(* (cbrt 1/3) (* (cbrt x) (cbrt (* 1/3 x))))

prune726.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.1b

regimes98.0ms

Accuracy

0% (0.2b remaining)

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

bsearch3.0ms

end0.0ms

sample15.5s

Algorithm
intervals
Results
9.0s3255×body2560valid
4.1s2486×body1280valid
1.6s1313×body640valid
333.0ms598×body320valid
234.0ms8000×pre80true
105.0ms299×body160valid
8.0ms49×body80valid