Average Error: 60.0 → 0.3
Time: 23.1s
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 2019018 
(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: 22.0s)Debug log

sample554.0ms

Algorithm
intervals
Results
378.0ms117×body2560valid
118.0ms79×body1280valid
26.0ms32×body640valid
13.0ms22×body320valid
8.0ms256×pre80true
2.0ms5×body160valid
0.0ms1×body80valid

simplify9.0ms

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

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.0b

localize19.0ms

Local error

Found 2 expressions with local error:

2.1b
(- (/ 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))

series138.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
110.0ms
(- (/ 1 x) (/ 1 (tan x)))
28.0ms
(/ 1 (tan x))

simplify316.0ms

Counts
20 → 41
Calls
20 calls:
Slowest
139.0ms
(- (/ 1 x) (+ (* 1/3 x) (* 1/45 (pow x 3))))
52.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
28.0ms
(/ (exp (/ 1 x)) (exp (/ 1 (tan x))))
12.0ms
(- (sqrt (/ 1 x)) (sqrt (/ 1 (tan x))))
9.0ms
(- (/ 1 x) (/ 1 (tan x)))

prune211.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize19.0ms

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)))

rewrite9.0ms

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
6.0ms
(+ (* 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)

series135.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
73.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
28.0ms
(* 1/45 (pow x 3))
26.0ms
(* 2/945 (pow x 5))
8.0ms
(* 1/3 x)

simplify125.0ms

Counts
20 → 55
Calls
20 calls:
Slowest
25.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
24.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
20.0ms
(* (exp (* 1/45 (pow x 3))) (exp (* 2/945 (pow x 5))))
14.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
5.0ms
(* 1/45 (pow x 3))

prune555.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

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

rewrite6.0ms

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
4.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
1.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt (* 1/3 x))

series1.2s

Counts
4 → 12
Calls
4 calls:
Slowest
350.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
303.0ms
(cbrt (* 1/3 x))
297.0ms
(cbrt (* 1/3 x))
296.0ms
(cbrt (* 1/3 x))

simplify165.0ms

Counts
31 → 63
Calls
31 calls:
Slowest
14.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
14.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
13.0ms
(* (* 1/3 x) (* 1/3 x))
13.0ms
(* (* 1/3 x) (* 1/3 x))
11.0ms
(* (cbrt -1/3) (pow (* x -1) 1/3))

prune688.0ms

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.2b

localize36.0ms

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 x)) (cbrt 1/3))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×pow1/3
4×add-log-exp
4×associate-*l*
4×add-cbrt-cube
3×associate-*r*
3×cbrt-prod
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
Counts
4 → 45
Calls
4 calls:
Slowest
8.0ms
(* (cbrt (* 1/3 x)) (cbrt 1/3))
1.0ms
(cbrt (* 1/3 x))
1.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt x)

series1.2s

Counts
4 → 12
Calls
4 calls:
Slowest
359.0ms
(* (cbrt (* 1/3 x)) (cbrt 1/3))
314.0ms
(cbrt (* 1/3 x))
269.0ms
(cbrt (* 1/3 x))
261.0ms
(cbrt x)

simplify95.0ms

Counts
24 → 57
Calls
24 calls:
Slowest
13.0ms
(* (pow (cbrt 1/3) 2) (pow x 1/3))
12.0ms
(* (* (cbrt -1/3) (cbrt 1/3)) (pow (* x -1) 1/3))
8.0ms
(* (pow (cbrt 1/3) 2) (pow x 1/3))
8.0ms
(* (* 1/3 x) 1/3)
6.0ms
(* (pow (* x -1) 1/3) (cbrt -1))

prune653.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.2b

regimes57.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample15.7s

Algorithm
intervals
Results
9.5s3317×body2560valid
4.3s2472×body1280valid
1.2s1250×body640valid
386.0ms623×body320valid
256.0ms8000×pre80true
105.0ms304×body160valid
6.0ms34×body80valid