Average Error: 0.2 → 0.2
Time: 38.9s
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

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

  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: 37.9s)Debug log

sample415.0ms

Algorithm
intervals

simplify114.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
113.0ms
(+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B)))

prune407.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize29.0ms

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)))

rewrite22.0ms

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
13.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
3.0ms
(/ x (tan B))
2.0ms
(/ 1 (sin B))

series302.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
170.0ms
(/ x (tan B))
105.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
26.0ms
(/ 1 (sin B))

simplify1.8s

Counts
66 → 85
Calls
66 calls:
Slowest
85.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
83.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
69.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
65.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
62.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (cos B) (/ x (sin B)))))

prune986.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

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)))

rewrite30.0ms

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
14.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
9.0ms
(/ (* x (cos B)) (sin B))
3.0ms
(* x (cos B))
2.0ms
(/ 1 (sin B))

series285.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
112.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
108.0ms
(/ (* x (cos B)) (sin B))
50.0ms
(* x (cos B))
15.0ms
(/ 1 (sin B))

simplify8.2s

Counts
101 → 117
Calls
101 calls:
Slowest
541.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
473.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
394.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
377.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (sin B)) (* x (cos B)))))
340.0ms
(fma (- (/ 1 (sin B))) (* x (cos B)) (* (/ 1 (sin B)) (* x (cos B))))

prune1.3s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize2.0ms

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)))

rewrite28.0ms

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
14.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
8.0ms
(/ (* x (cos B)) (sin B))
2.0ms
(* x (cos B))
2.0ms
(/ 1 (sin B))

series332.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
153.0ms
(/ (* x (cos B)) (sin B))
107.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
49.0ms
(* x (cos B))
23.0ms
(/ 1 (sin B))

simplify8.4s

Counts
101 → 117
Calls
101 calls:
Slowest
580.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
493.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
411.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
342.0ms
(fma (- (/ 1 (sin B))) (* x (cos B)) (* (/ 1 (sin B)) (* x (cos B))))
341.0ms
(fma (- (/ 1 (sin B))) (* x (cos B)) (* (/ 1 (sin B)) (* x (cos B))))

prune1.2s

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)))

rewrite13.0ms

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
6.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))
3.0ms
(/ (tan B) x)
1.0ms
(/ 1 (/ (tan B) x))
1.0ms
(/ 1 (sin B))

series327.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
151.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))
82.0ms
(/ 1 (/ (tan B) x))
70.0ms
(/ (tan B) x)
24.0ms
(/ 1 (sin B))

simplify3.3s

Counts
75 → 110
Calls
75 calls:
Slowest
110.0ms
(fma 1 (/ 1 (sin B)) (- (* x (/ 1 (tan B)))))
100.0ms
(fma (- (/ 1 (/ (tan B) x))) 1 (* (/ 1 (/ (tan B) x)) 1))
95.0ms
(fma (- (/ 1 (/ (tan B) x))) 1 (* (/ 1 (/ (tan B) x)) 1))
88.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (/ (tan B) x)) 1)))
86.0ms
(fma (- (/ 1 (/ (tan B) x))) 1 (* (/ 1 (/ (tan B) x)) 1))

prune1.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

regimes125.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample9.3s

Algorithm
intervals