Average Error: 31.8 → 10.1
Time: 4.1m
Precision: 64
Internal Precision: 128
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -16394960.105851:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;y.re \le 4.7496459374994184 \cdot 10^{-210} \lor \neg \left(y.re \le 1.2409494224095134 \cdot 10^{-148}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if y.re < -16394960.105851

    1. Initial program 38.2

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around -inf 0.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    3. Simplified0.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    if -16394960.105851 < y.re < 4.7496459374994184e-210 or 1.2409494224095134e-148 < y.re

    1. Initial program 25.5

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around -inf 29.4

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    3. Simplified29.4

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    4. Taylor expanded around -inf 18.3

      \[\leadsto e^{\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    5. Simplified18.3

      \[\leadsto e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    if 4.7496459374994184e-210 < y.re < 1.2409494224095134e-148

    1. Initial program 30.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -16394960.105851:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;y.re \le 4.7496459374994184 \cdot 10^{-210} \lor \neg \left(y.re \le 1.2409494224095134 \cdot 10^{-148}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))

Details

Time bar (total: 4.0m)Debug log

sample7.9s

Algorithm
intervals

simplify87.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
86.0ms
(* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re))))

prune23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 33.4b

localize59.0ms

Local error

Found 4 expressions with local error:

32.7b
(sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))
31.6b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
31.6b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
0.3b
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)

rewrite33.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×sqrt-prod
5×associate-*l*
5×pow1
4×add-log-exp
4×add-exp-log
4×sqrt-div
4×add-cbrt-cube
3×pow1/2
2×flip-+
2×log-pow
2×flip3-+
2×rem-sqrt-square
1×*-commutative
1×sin-sum
Counts
4 → 49
Calls
4 calls:
Slowest
18.0ms
(sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))
7.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
3.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
2.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

series297.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
192.0ms
(sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))
67.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
19.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
18.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

simplify585.0ms

Counts
27 → 61
Calls
27 calls:
Slowest
112.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
106.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
69.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
62.0ms
(* -1 (* y.re (log (/ -1 x.re))))
58.0ms
(* -1 (* y.re (log (/ 1 x.re))))

prune1.1s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 13.3b

localize18.0ms

Local error

Found 4 expressions with local error:

32.8b
(sin (+ (* (log (- x.re)) y.im) (* (atan2 x.im x.re) y.re)))
31.6b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
0.3b
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
0.3b
(* (log (- x.re)) y.im)

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
9×associate-*l*
8×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×pow1
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
3×sqrt-prod
3×log-pow
2×*-commutative
2×sqrt-div
2×pow1/2
1×flip-+
1×sin-sum
1×flip3-+
1×rem-sqrt-square
Counts
4 → 47
Calls
4 calls:
Slowest
10.0ms
(sin (+ (* (log (- x.re)) y.im) (* (atan2 x.im x.re) y.re)))
3.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
3.0ms
(* (log (- x.re)) y.im)
2.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

series311.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
107.0ms
(sin (+ (* (log (- x.re)) y.im) (* (atan2 x.im x.re) y.re)))
95.0ms
(* (log (- x.re)) y.im)
88.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
22.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

simplify615.0ms

Counts
23 → 59
Calls
23 calls:
Slowest
127.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
80.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
77.0ms
(sin (- (+ (* (atan2 x.im x.re) y.re) (* y.im (log -1))) (* y.im (log (/ 1 x.re)))))
63.0ms
(* -1 (* y.im (log (/ -1 x.re))))
59.0ms
(* -1 (* y.re (log (/ 1 x.re))))

prune1.2s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 6.5b

localize13.0ms

Local error

Found 4 expressions with local error:

32.8b
(sin (+ (* (log (- x.re)) y.im) (* (atan2 x.im x.re) y.re)))
0.3b
(* (log (- x.re)) y.re)
0.3b
(* (log (- x.re)) y.im)
0.2b
(* (atan2 x.im x.re) y.re)

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
11×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×pow1
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
3×*-commutative
2×log-pow
1×sin-sum
Counts
4 → 43
Calls
4 calls:
Slowest
14.0ms
(sin (+ (* (log (- x.re)) y.im) (* (atan2 x.im x.re) y.re)))
3.0ms
(* (log (- x.re)) y.re)
3.0ms
(* (log (- x.re)) y.im)
2.0ms
(* (atan2 x.im x.re) y.re)

series331.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
133.0ms
(sin (+ (* (log (- x.re)) y.im) (* (atan2 x.im x.re) y.re)))
104.0ms
(* (log (- x.re)) y.im)
82.0ms
(* (log (- x.re)) y.re)
12.0ms
(* (atan2 x.im x.re) y.re)

simplify480.0ms

Counts
19 → 55
Calls
19 calls:
Slowest
101.0ms
(sin (- (+ (* (atan2 x.im x.re) y.re) (* y.im (log -1))) (* y.im (log (/ 1 x.re)))))
89.0ms
(* -1 (* y.re (log (/ -1 x.re))))
87.0ms
(* -1 (* y.im (log (/ -1 x.re))))
62.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
41.0ms
(* y.re (- (log -1) (log (/ 1 x.re))))

prune975.0ms

Pruning

14 alts after pruning (11 fresh and 3 done)

Merged error: 6.4b

localize31.0ms

Local error

Found 4 expressions with local error:

32.8b
(sin (+ (* (* (cbrt (log (- x.re))) (cbrt (log (- x.re)))) (* (cbrt (log (- x.re))) y.im)) (* (atan2 x.im x.re) y.re)))
0.5b
(* (cbrt (log (- x.re))) (cbrt (log (- x.re))))
0.5b
(cbrt (log (- x.re)))
0.5b
(cbrt (log (- x.re)))

rewrite72.0ms

Algorithm
rewrite-expression-head
Rules
16×cbrt-prod
13×pow1
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
7×associate-*l*
7×associate-*r*
6×add-exp-log
6×pow1/3
4×add-log-exp
4×log-pow
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sin-sum
1×pow-plus
1×pow2
Counts
4 → 62
Calls
4 calls:
Slowest
62.0ms
(sin (+ (* (* (cbrt (log (- x.re))) (cbrt (log (- x.re)))) (* (cbrt (log (- x.re))) y.im)) (* (atan2 x.im x.re) y.re)))
7.0ms
(* (cbrt (log (- x.re))) (cbrt (log (- x.re))))
1.0ms
(cbrt (log (- x.re)))
1.0ms
(cbrt (log (- x.re)))

series1.3s

Counts
4 → 12
Calls
4 calls:
Slowest
399.0ms
(* (cbrt (log (- x.re))) (cbrt (log (- x.re))))
385.0ms
(cbrt (log (- x.re)))
376.0ms
(cbrt (log (- x.re)))
144.0ms
(sin (+ (* (* (cbrt (log (- x.re))) (cbrt (log (- x.re)))) (* (cbrt (log (- x.re))) y.im)) (* (atan2 x.im x.re) y.re)))

simplify442.0ms

Counts
43 → 74
Calls
43 calls:
Slowest
85.0ms
(sin (- (+ (* (atan2 x.im x.re) y.re) (* y.im (log -1))) (* y.im (log (/ 1 x.re)))))
78.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
61.0ms
(pow (pow (- (log -1) (log (/ 1 x.re))) 2) 1/3)
38.0ms
(+ (* y.im (log -1)) (+ (* (atan2 x.im x.re) y.re) (* y.im (log x.re))))
27.0ms
(* (sin (* (* (cbrt (log (- x.re))) (cbrt (log (- x.re)))) (* (cbrt (log (- x.re))) y.im))) (cos (* (atan2 x.im x.re) y.re)))

prune1.3s

Pruning

15 alts after pruning (12 fresh and 3 done)

Merged error: 6.4b

regimes979.0ms

Accuracy

52.1% (5.4b remaining)

Error of 10.1b against oracle of 4.7b and baseline of 16.0b

bsearch14.6s

end0.0ms

sample3.5m

Algorithm
intervals