Average Error: 59.9 → 0.3
Time: 36.0s
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 323378853 
(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: 32.7s)Debug log

sample685.0ms

Algorithm
intervals
Results
418.0ms104×body2560valid
168.0ms84×body1280valid
51.0ms39×body640valid
20.0ms20×body320valid
10.0ms256×pre80true
6.0ms9×body160valid

simplify10.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
10.0ms
(- (/ 1 x) (/ 1 (tan x)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.2b

localize29.0ms

Local error

Found 1 expressions with local error:

2.5b
(- (/ 1 x) (/ 1 (tan x)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
5×*-un-lft-identity
4×distribute-lft-out--
4×div-inv
3×add-log-exp
3×add-sqr-sqrt
1×difference-of-squares
1×flip--
1×add-cube-cbrt
1×frac-sub
1×add-exp-log
1×diff-log
1×add-cbrt-cube
1×flip3--
1×sub-neg
1×pow1
Counts
1 → 18
Calls
1 calls:
Slowest
10.0ms
(- (/ 1 x) (/ 1 (tan x)))

series128.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
128.0ms
(- (/ 1 x) (/ 1 (tan x)))

simplify186.0ms

Counts
12 → 21
Calls
12 calls:
Slowest
62.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
36.0ms
(/ (exp (/ 1 x)) (exp (/ 1 (tan x))))
10.0ms
(- (sqrt (/ 1 x)) (sqrt (/ 1 (tan x))))
10.0ms
(- (* 1 (tan x)) (* x 1))
10.0ms
(- (/ 1 x) (/ (cos x) (sin x)))

prune250.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize53.0ms

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)))

rewrite15.0ms

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
8.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
3.0ms
(* 1/45 (pow x 3))
3.0ms
(* 2/945 (pow x 5))
0.0ms
(* 1/3 x)

series206.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
110.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
46.0ms
(* 2/945 (pow x 5))
32.0ms
(* 1/45 (pow x 3))
18.0ms
(* 1/3 x)

simplify191.0ms

Counts
20 → 55
Calls
20 calls:
Slowest
40.0ms
(* (exp (* 1/45 (pow x 3))) (exp (* 2/945 (pow x 5))))
27.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
27.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
26.0ms
(+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5)))
11.0ms
(* 1/45 (pow x 3))

prune878.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.3b

localize32.0ms

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)))

rewrite14.0ms

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
10.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
1.0ms
(cbrt (* 1/3 x))
1.0ms
(cbrt (* 1/3 x))
1.0ms
(cbrt (* 1/3 x))

series2.0s

Counts
4 → 12
Calls
4 calls:
Slowest
576.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
491.0ms
(cbrt (* 1/3 x))
480.0ms
(cbrt (* 1/3 x))
474.0ms
(cbrt (* 1/3 x))

simplify210.0ms

Counts
31 → 63
Calls
31 calls:
Slowest
20.0ms
(* (pow (cbrt -1/3) 2) (pow (pow x 2) 1/3))
18.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
18.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
18.0ms
(* (* 1/3 x) (* 1/3 x))
17.0ms
(* (* 1/3 x) (* 1/3 x))

prune1.2s

Pruning

11 alts after pruning (9 fresh and 2 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))
0.5b
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))

rewrite14.0ms

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
4×add-log-exp
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
4×cbrt-prod
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 50
Calls
4 calls:
Slowest
10.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
1.0ms
(cbrt (* 1/3 x))
1.0ms
(cbrt (* 1/3 x))
0.0ms
(cbrt x)

series2.0s

Counts
4 → 12
Calls
4 calls:
Slowest
571.0ms
(* (cbrt (* 1/3 x)) (cbrt (* 1/3 x)))
478.0ms
(cbrt (* 1/3 x))
473.0ms
(cbrt (* 1/3 x))
443.0ms
(cbrt x)

simplify201.0ms

Counts
30 → 62
Calls
30 calls:
Slowest
20.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
19.0ms
(* (pow (cbrt -1/3) 2) (pow (pow x 2) 1/3))
18.0ms
(* (pow (cbrt 1/3) 2) (pow (pow x 2) 1/3))
17.0ms
(* (* 1/3 x) (* 1/3 x))
16.0ms
(* (* 1/3 x) (* 1/3 x))

prune1.3s

Pruning

16 alts after pruning (14 fresh and 2 done)

Merged error: 0.2b

regimes196.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample22.9s

Algorithm
intervals
Results
13.8s3226×body2560valid
5.9s2513×body1280valid
1.8s1249×body640valid
625.0ms632×body320valid
366.0ms8000×pre80true
276.0ms324×body160valid
18.0ms56×body80valid