Average Error: 0.2 → 0.2
Time: 37.7s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]

  3. Taylor expanded around -inf 0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}}\]

  4. Taylor expanded around inf 0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x \cdot \cos B}{\sin B}\]

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Details

Time bar (total: 36.7s)Debug log

sample160.0ms

Algorithm
intervals

simplify74.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
74.0ms
(+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B)))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize20.0ms

Local error

Found 3 expressions with local error:

0.2b
(/ x (tan B))
0.2b
(/ 1 (sin B))
0.0b
(- (/ 1 (sin B)) (/ x (tan B)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
20×prod-diff
18×*-un-lft-identity
17×add-sqr-sqrt
15×add-cube-cbrt
13×div-inv
6×associate-/r*
5×add-log-exp
5×associate-/r/
5×tan-quot
4×add-exp-log
4×fma-neg
4×pow1
3×log1p-expm1-u
3×add-cbrt-cube
3×expm1-log1p-u
2×distribute-lft-out--
2×frac-2neg
2×clear-num
1×difference-of-squares
1×inv-pow
1×flip--
1×pow-flip
1×frac-sub
1×diff-log
1×flip3--
1×sub-neg
1×rec-exp
Counts
3 → 76
Calls
3 calls:
Slowest
7.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
1.0ms
(/ 1 (sin B))
1.0ms
(/ x (tan B))

series265.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
128.0ms
(/ x (tan B))
115.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
22.0ms
(/ 1 (sin B))

simplify1.3s

Counts
66 → 85
Calls
66 calls:
Slowest
50.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
50.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
49.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
44.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
41.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))

prune598.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (* x (cos B)) (sin B))
0.2b
(/ 1 (sin B))
0.1b
(* x (cos B))
0.0b
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
28×prod-diff
28×*-un-lft-identity
26×add-sqr-sqrt
24×add-cube-cbrt
18×div-inv
15×times-frac
7×add-exp-log
6×add-log-exp
6×add-cbrt-cube
6×associate-/r*
5×pow1
4×log1p-expm1-u
4×fma-neg
4×expm1-log1p-u
3×associate-*r*
2×distribute-lft-out--
2×distribute-rgt-out--
2×frac-2neg
2×clear-num
1×difference-of-squares
1×sub-div
1×inv-pow
1×associate-/l*
1×flip--
1×pow-flip
1×frac-sub
1×*-commutative
1×div-exp
1×diff-log
1×flip3--
1×sub-neg
1×rec-exp
1×cbrt-undiv
Counts
4 → 105
Calls
4 calls:
Slowest
7.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
4.0ms
(/ (* x (cos B)) (sin B))
1.0ms
(* x (cos B))
1.0ms
(/ 1 (sin B))

series248.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
98.0ms
(/ (* x (cos B)) (sin B))
96.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
39.0ms
(* x (cos B))
14.0ms
(/ 1 (sin B))

simplify9.2s

Counts
101 → 117
Calls
101 calls:
Slowest
544.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
470.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (sin B)) (* x (cos B)))))
420.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
372.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
346.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ (cos B) (sqrt (sin B))) (/ x (sqrt (sin B))))))

prune1.2s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize1.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (* x (cos B)) (sin B))
0.2b
(/ 1 (sin B))
0.1b
(* x (cos B))
0.0b
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
28×prod-diff
28×*-un-lft-identity
26×add-sqr-sqrt
24×add-cube-cbrt
18×div-inv
15×times-frac
7×add-exp-log
6×add-log-exp
6×add-cbrt-cube
6×associate-/r*
5×pow1
4×log1p-expm1-u
4×fma-neg
4×expm1-log1p-u
3×associate-*r*
2×distribute-lft-out--
2×distribute-rgt-out--
2×frac-2neg
2×clear-num
1×difference-of-squares
1×sub-div
1×inv-pow
1×associate-/l*
1×flip--
1×pow-flip
1×frac-sub
1×*-commutative
1×div-exp
1×diff-log
1×flip3--
1×sub-neg
1×rec-exp
1×cbrt-undiv
Counts
4 → 105
Calls
4 calls:
Slowest
9.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
4.0ms
(/ (* x (cos B)) (sin B))
1.0ms
(* x (cos B))
1.0ms
(/ 1 (sin B))

series295.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
122.0ms
(/ (* x (cos B)) (sin B))
112.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
35.0ms
(* x (cos B))
26.0ms
(/ 1 (sin B))

simplify8.6s

Counts
101 → 117
Calls
101 calls:
Slowest
532.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
466.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
410.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
383.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (sin B)) (* x (cos B)))))
331.0ms
(fma (- (/ 1 (sin B))) (* x (cos B)) (* (/ 1 (sin B)) (* x (cos B))))

prune1.1s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ 1 (/ (tan B) x))
0.2b
(/ (tan B) x)
0.2b
(/ 1 (sin B))
0.0b
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
21×*-un-lft-identity
20×prod-diff
19×add-sqr-sqrt
18×div-inv
17×add-cube-cbrt
7×associate-/r*
6×add-log-exp
6×add-exp-log
6×pow1
5×associate-/r/
4×distribute-lft-out--
4×log1p-expm1-u
4×fma-neg
4×add-cbrt-cube
4×expm1-log1p-u
3×associate-/l*
3×frac-2neg
3×clear-num
2×inv-pow
2×pow-flip
2×rec-exp
1×associate-/l/
1×difference-of-squares
1×flip--
1×frac-sub
1×tan-quot
1×diff-log
1×flip3--
1×sub-neg
Counts
4 → 98
Calls
4 calls:
Slowest
12.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))
3.0ms
(/ (tan B) x)
1.0ms
(/ 1 (/ (tan B) x))
1.0ms
(/ 1 (sin B))

series297.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
121.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))
106.0ms
(/ 1 (/ (tan B) x))
45.0ms
(/ (tan B) x)
25.0ms
(/ 1 (sin B))

simplify3.1s

Counts
75 → 110
Calls
75 calls:
Slowest
132.0ms
(+ (/ B x) (+ (* 1/3 (/ (pow B 3) x)) (* 2/15 (/ (pow B 5) x))))
100.0ms
(fma (- (/ 1 (/ (tan B) x))) 1 (* (/ 1 (/ (tan B) x)) 1))
99.0ms
(fma (- (/ 1 (/ (tan B) x))) 1 (* (/ 1 (/ (tan B) x)) 1))
96.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (/ (tan B) x)) 1)))
88.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* x (/ 1 (tan B)))))

prune1.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

regimes109.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch1.0ms

end0.0ms

sample8.9s

Algorithm
intervals