Average Error: 60.0 → 0.3
Time: 34.4s
Precision: 64
Internal Precision: 128
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[(\left((\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*\right) \cdot x + \left(\frac{2}{945} \cdot {x}^{5}\right))_*\]

Error

Bits error versus x

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. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(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: 32.9s)Debug log

sample461.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.0b

localize11.0ms

Local error

Found 2 expressions with local error:

3.2b
(- (/ 1 x) (/ 1 (tan x)))
0.0b
(/ 1 (tan x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 63
Calls

2 calls. Slowest were:

5.0ms
(- (/ 1 x) (/ 1 (tan x)))
1.0ms
(/ 1 (tan x))

series146.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

109.0ms
(- (/ 1 x) (/ 1 (tan x)))
37.0ms
(/ 1 (tan x))

simplify1.6s

Counts
62 → 69
Calls

62 calls. Slowest were:

209.0ms
(- (/ 1 x) (+ (* 1/3 x) (* 1/45 (pow x 3))))
103.0ms
(fma (- (cos x)) (/ 1 (sin x)) (* (cos x) (/ 1 (sin x))))
96.0ms
(fma 1 (/ 1 x) (- (* (cos x) (/ 1 (sin x)))))

prune539.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize38.0ms

Local error

Found 3 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

5.0ms
(* (pow x 5) 2/945)
0.0ms
(fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))
0.0ms
(fma 1/45 (* x x) 1/3)

series108.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

38.0ms
(fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))
35.0ms
(* (pow x 5) 2/945)
34.0ms
(fma 1/45 (* x x) 1/3)

simplify250.0ms

Counts
11 → 42
Calls

11 calls. Slowest were:

88.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
70.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
45.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))

prune417.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.2b

localize11.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
0.7b
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
0.7b
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
0.5b
(* (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))) (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))) (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))))
1.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
1.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))

series568.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

154.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
148.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
139.0ms
(* (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))) (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))))
127.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))

simplify5.0s

Counts
41 → 81
Calls

41 calls. Slowest were:

424.0ms
(+ (* 7/2 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 2))) (+ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (* 161/4 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 4)))))
400.0ms
(+ (* 7/2 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 2))) (+ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (* 161/4 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 4)))))
376.0ms
(+ (exp (* 1/3 (- (log 4/893025) (* 10 (log (/ 1 x)))))) (+ (* 371/4 (/ (exp (* 1/3 (- (log 4/893025) (* 10 (log (/ 1 x)))))) (pow x 4))) (* 7 (/ (exp (* 1/3 (- (log 4/893025) (* 10 (log (/ 1 x)))))) (pow x 2)))))

prune956.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.2b

localize3.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
0.7b
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
0.7b
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
0.5b
(* (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))) (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))) (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))))
1.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
1.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))

series620.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

193.0ms
(* (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))) (cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))))
152.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
140.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))
135.0ms
(cbrt (fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945)))

simplify5.2s

Counts
41 → 81
Calls

41 calls. Slowest were:

484.0ms
(+ (* 7/2 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 2))) (+ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (* 161/4 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 4)))))
422.0ms
(+ (* 7/2 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 2))) (+ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (* 161/4 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 4)))))
377.0ms
(+ (exp (* 1/3 (- (log 4/893025) (* 10 (log (/ 1 x)))))) (+ (* 371/4 (/ (exp (* 1/3 (- (log 4/893025) (* 10 (log (/ 1 x)))))) (pow x 4))) (* 7 (/ (exp (* 1/3 (- (log 4/893025) (* 10 (log (/ 1 x)))))) (pow x 2)))))

prune969.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.2b

regimes87.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample15.9s

Algorithm
intervals