Average Error: 59.9 → 0.3
Time: 39.9s
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

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. Initial simplification59.9

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. 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 2018362 +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: 14.9s)Debug log

start728.0ms

Algorithm
intervals

setup23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.2b

localize22.0ms

Local error

Found 2 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 63
Calls

2 calls. Slowest were:

9.0ms
(- (/ 1 x) (/ 1 (tan x)))
2.0ms
(/ 1 (tan x))

series128.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

97.0ms
(- (/ 1 x) (/ 1 (tan x)))
31.0ms
(/ 1 (tan x))

simplify2.1s

Counts
62 → 69
Calls

62 calls. Slowest were:

267.0ms
(- (/ 1 x) (+ (* 1/3 x) (* 1/45 (pow x 3))))
127.0ms
(fma (- (cos x)) (/ 1 (sin x)) (* (cos x) (/ 1 (sin x))))
115.0ms
(fma (- (cos x)) (/ 1 (sin x)) (* (cos x) (/ 1 (sin x))))

prune557.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize16.0ms

Local error

Found 3 expressions with local error:

0.4b
(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)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

2.0ms
(* (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)

series92.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

41.0ms
(fma 1/45 (* x x) 1/3)
28.0ms
(* (pow x 5) 2/945)
23.0ms
(fma (fma 1/45 (* x x) 1/3) x (* (pow x 5) 2/945))

simplify606.0ms

Counts
11 → 42
Calls

11 calls. Slowest were:

433.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
73.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))
58.0ms
(+ (* 1/3 x) (+ (* 1/45 (pow x 3)) (* 2/945 (pow x 5))))

prune578.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.3b

localize10.0ms

Local error

Found 4 expressions with local error:

0.4b
(fma (expm1 (log1p (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)
0.0b
(log1p (fma 1/45 (* x x) 1/3))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

2.0ms
(* (pow x 5) 2/945)
0.0ms
(log1p (fma 1/45 (* x x) 1/3))
0.0ms
(fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))

series285.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))
102.0ms
(log1p (fma 1/45 (* x x) 1/3))
39.0ms
(fma 1/45 (* x x) 1/3)
33.0ms
(* (pow x 5) 2/945)

simplify1.3s

Counts
15 → 56
Calls

15 calls. Slowest were:

343.0ms
(- (+ (* 60 (/ (exp (- (log 1/45) (* 2 (log (/ 1 x))))) x)) (+ (* 2/945 (pow x 5)) (* x (exp (- (log 1/45) (* 2 (log (/ 1 x)))))))) x)
319.0ms
(- (+ (* x (exp (- (log 1/45) (* 2 (log (/ -1 x)))))) (+ (* 60 (/ (exp (- (log 1/45) (* 2 (log (/ -1 x))))) x)) (* 2/945 (pow x 5)))) x)
236.0ms
(- (+ (log 1/45) (* 60 (/ 1 (pow x 2)))) (+ (* 2 (log (/ -1 x))) (* 1800 (/ 1 (pow x 4)))))

prune794.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.2b

localize12.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))
0.7b
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))
0.7b
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))
0.5b
(* (cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))) (cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

7.0ms
(* (cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))) (cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))))
1.0ms
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))
1.0ms
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

383.0ms
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))
381.0ms
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))
308.0ms
(* (cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))) (cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945))))
288.0ms
(cbrt (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))

simplify4.8s

Counts
41 → 81
Calls

41 calls. Slowest were:

328.0ms
(+ (exp (* 1/3 (+ (log 1/3) (log x)))) (+ (* 23/14175 (* (pow x 4) (exp (* 1/3 (+ (log 1/3) (log x)))))) (* 1/45 (* (pow x 2) (exp (* 1/3 (+ (log 1/3) (log x))))))))
317.0ms
(- (+ (* 315/2 (/ (* (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (exp (- (log 1/45) (* 2 (log (/ 1 x)))))) (pow x 4))) (+ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (* 9450 (/ (* (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (exp (- (log 1/45) (* 2 (log (/ 1 x)))))) (pow x 6))))) (* 315/2 (/ (exp (* 1/3 (- (log 2/945) (* 5 (log (/ 1 x)))))) (pow x 4))))
308.0ms
(* (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)) (fma (expm1 (log1p (fma 1/45 (* x x) 1/3))) x (* (pow x 5) 2/945)))

prune1.4s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.2b

regimes65.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms