Average Error: 30.7 → 0.2
Time: 2.4m
Precision: 64
Internal Precision: 128
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -7.152739789478762 \cdot 10^{-15}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

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 2 regimes

  2. if y.re < -7.152739789478762e-15

    1. Initial program 36.8

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

    2. Taylor expanded around 0 0.3

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

    if -7.152739789478762e-15 < y.re

    1. Initial program 27.5

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

    2. Taylor expanded around 0 25.3

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

      \[\leadsto e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -7.152739789478762 \cdot 10^{-15}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))

Details

Time bar (total: 2.4m)Debug log

sample4.3s

Algorithm
intervals
Results
1234×(pre true 80)
977×(body exit 10240)
257×(body real 80)

simplify458.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
458.0ms
(* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re))))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.0b

localize43.0ms

Local error

Found 4 expressions with local error:

27.7b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
27.7b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
27.3b
(cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))
0.3b
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)

rewrite47.0ms

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×cos-sum
Counts
4 → 49
Calls
4 calls:
Slowest
29.0ms
(cos (+ (* (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)
4.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
4.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

series286.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
144.0ms
(cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))
81.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
31.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
30.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

simplify668.0ms

Counts
27 → 61
Calls
27 calls:
Slowest
150.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
118.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
81.0ms
(cos (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
81.0ms
(* -1 (* y.re (log (/ 1 x.re))))
76.0ms
(* -1 (* y.re (log (/ -1 x.re))))

prune1.1s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.7b

localize8.0ms

Local error

Found 4 expressions with local error:

27.7b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
0.3b
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
0.2b
(* (atan2 x.im x.re) y.im)
0.0b
(- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*l*
8×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×add-log-exp
5×pow1
4×add-exp-log
4×add-cbrt-cube
3×sqrt-prod
2×*-commutative
2×log-pow
2×sqrt-div
2×pow1/2
1×flip--
1×flip-+
1×diff-log
1×flip3--
1×flip3-+
1×sub-neg
1×rem-sqrt-square
Counts
4 → 50
Calls
4 calls:
Slowest
12.0ms
(- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))
6.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
5.0ms
(* (atan2 x.im x.re) y.im)
4.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

series273.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
134.0ms
(- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))
86.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
32.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
22.0ms
(* (atan2 x.im x.re) y.im)

simplify501.0ms

Counts
23 → 62
Calls
23 calls:
Slowest
107.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
101.0ms
(- (+ (* y.re (log (/ -1 x.re))) (* y.im (atan2 x.im x.re))))
73.0ms
(* -1 (* y.re (log (/ -1 x.re))))
63.0ms
(* -1 (* y.re (log (/ 1 x.re))))
38.0ms
(/ (exp (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)) (exp (* (atan2 x.im x.re) y.im)))

prune847.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (log (- x.re)) y.re)
0.2b
(* (atan2 x.im x.re) y.im)
0.0b
(exp (- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im)))
0.0b
(- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
9×add-log-exp
7×add-cube-cbrt
7×associate-*l*
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow1
4×add-exp-log
4×add-cbrt-cube
3×exp-prod
2×rem-exp-log
2×*-commutative
2×exp-sum
2×diff-log
2×sub-neg
1×flip--
1×log-pow
1×exp-diff
1×flip3--
Counts
4 → 50
Calls
4 calls:
Slowest
9.0ms
(exp (- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im)))
4.0ms
(- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im))
3.0ms
(* (log (- x.re)) y.re)
2.0ms
(* (atan2 x.im x.re) y.im)

series322.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
117.0ms
(- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im))
102.0ms
(* (log (- x.re)) y.re)
89.0ms
(exp (- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im)))
13.0ms
(* (atan2 x.im x.re) y.im)

simplify1.1s

Counts
25 → 62
Calls
25 calls:
Slowest
226.0ms
(exp (- (+ (* y.re (log (/ -1 x.re))) (* y.im (atan2 x.im x.re)))))
172.0ms
(exp (- (* y.re (log -1)) (+ (* y.im (atan2 x.im x.re)) (* y.re (log (/ 1 x.re))))))
159.0ms
(- (+ (* y.re (log -1)) (+ (* y.re (log x.re)) 1)) (* y.im (atan2 x.im x.re)))
129.0ms
(- (+ (* y.re (log (/ -1 x.re))) (* y.im (atan2 x.im x.re))))
96.0ms
(* -1 (* y.re (log (/ -1 x.re))))

prune846.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (atan2 x.im x.re) y.im)
0.2b
(* y.re (log x.im))
0.0b
(- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×add-exp-log
3×associate-*l*
3×associate-*r*
3×add-cbrt-cube
3×pow1
2×*-commutative
1×flip--
1×diff-log
1×flip3--
1×sub-neg
Counts
3 → 34
Calls
3 calls:
Slowest
3.0ms
(- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im))
2.0ms
(* (atan2 x.im x.re) y.im)
1.0ms
(* y.re (log x.im))

series129.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
76.0ms
(- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im))
39.0ms
(* y.re (log x.im))
13.0ms
(* (atan2 x.im x.re) y.im)

simplify502.0ms

Counts
15 → 43
Calls
15 calls:
Slowest
212.0ms
(- (* y.re (log -1)) (+ (* y.re (log (/ -1 x.im))) (* y.im (atan2 x.im x.re))))
103.0ms
(* y.re (- (log -1) (log (/ -1 x.im))))
76.0ms
(* -1 (* y.re (log (/ 1 x.im))))
49.0ms
(- (+ (* y.re (log (/ 1 x.im))) (* y.im (atan2 x.im x.re))))
35.0ms
(/ (exp (* y.re (log x.im))) (exp (* (atan2 x.im x.re) y.im)))

prune463.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes339.0ms

Accuracy

97.8% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 7.6b

bsearch2.8s

end0.0ms

sample2.2m

Algorithm
intervals
Results
38786×(pre true 80)
30786×(body exit 10240)
8000×(body real 80)