Average Error: 31.0 → 0.3
Time: 2.3m
Precision: 64
Internal Precision: 128
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -9.036063513144656 \cdot 10^{-18}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

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 < -9.036063513144656e-18

    1. Initial program 37.4

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

    2. Taylor expanded around 0 0.8

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

    if -9.036063513144656e-18 < y.re

    1. Initial program 27.6

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

    2. Taylor expanded around 0 25.2

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -9.036063513144656 \cdot 10^{-18}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(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: 2.3m)Debug log

sample4.3s

Algorithm
intervals
Results
4.1s1022×body10240exit
96.0ms256×body80valid

simplify120.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
120.0ms
(* (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))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.7b

localize56.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
28.2b
(sqrt (+ (* x.re x.re) (* x.im x.im)))
27.9b
(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.im)

rewrite32.0ms

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
17.0ms
(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.im)
4.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
2.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

series312.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
157.0ms
(cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))
90.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im)
40.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
25.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))

simplify605.0ms

Counts
27 → 61
Calls
27 calls:
Slowest
124.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
105.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
75.0ms
(cos (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
75.0ms
(* -1 (* y.im (log (/ 1 x.re))))
63.0ms
(* -1 (* y.im (log (/ -1 x.re))))

prune1.1s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.6b

localize7.0ms

Local error

Found 4 expressions with local error:

28.2b
(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))

rewrite25.0ms

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
9.0ms
(- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))
7.0ms
(* (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)))

series275.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
129.0ms
(- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))
101.0ms
(* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
30.0ms
(sqrt (+ (* x.re x.re) (* x.im x.im)))
15.0ms
(* (atan2 x.im x.re) y.im)

simplify547.0ms

Counts
23 → 62
Calls
23 calls:
Slowest
165.0ms
(sqrt (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))))
102.0ms
(- (+ (* y.re (log (/ -1 x.re))) (* y.im (atan2 x.im x.re))))
66.0ms
(* -1 (* y.re (log (/ -1 x.re))))
65.0ms
(* -1 (* y.re (log (/ 1 x.re))))
34.0ms
(sqrt (+ (pow (* x.re x.re) 3) (pow (* x.im x.im) 3)))

prune910.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize30.0ms

Local error

Found 3 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
7×associate-*l*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×pow1
3×add-exp-log
3×add-cbrt-cube
2×*-commutative
1×flip--
1×log-pow
1×diff-log
1×flip3--
1×sub-neg
Counts
3 → 35
Calls
3 calls:
Slowest
9.0ms
(- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im))
6.0ms
(* (log (- x.re)) y.re)
5.0ms
(* (atan2 x.im x.re) y.im)

series211.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
113.0ms
(- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im))
84.0ms
(* (log (- x.re)) y.re)
15.0ms
(* (atan2 x.im x.re) y.im)

simplify449.0ms

Counts
14 → 44
Calls
14 calls:
Slowest
121.0ms
(- (+ (* y.re (log (/ -1 x.re))) (* y.im (atan2 x.im x.re))))
94.0ms
(* -1 (* y.re (log (/ -1 x.re))))
93.0ms
(- (* y.re (log -1)) (+ (* y.im (atan2 x.im x.re)) (* y.re (log (/ 1 x.re)))))
63.0ms
(- (+ (* y.re (log -1)) (* y.re (log x.re))) (* y.im (atan2 x.im x.re)))
29.0ms
(* y.re (- (log -1) (log (/ 1 x.re))))

prune575.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize20.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (atan2 x.im x.re) y.im)
0.1b
(* (log x.im) y.re)
0.0b
(- (* (log x.im) y.re) (* (atan2 x.im x.re) y.im))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
6×associate-*l*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×add-exp-log
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
9.0ms
(- (* (log x.im) y.re) (* (atan2 x.im x.re) y.im))
5.0ms
(* (atan2 x.im x.re) y.im)
5.0ms
(* (log x.im) y.re)

series196.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
102.0ms
(- (* (log x.im) y.re) (* (atan2 x.im x.re) y.im))
73.0ms
(* (log x.im) y.re)
21.0ms
(* (atan2 x.im x.re) y.im)

simplify442.0ms

Counts
14 → 43
Calls
14 calls:
Slowest
225.0ms
(- (* y.re (log -1)) (+ (* y.re (log (/ -1 x.im))) (* y.im (atan2 x.im x.re))))
62.0ms
(* -1 (* y.re (log (/ 1 x.im))))
53.0ms
(* y.re (- (log -1) (log (/ -1 x.im))))
45.0ms
(- (+ (* y.re (log (/ 1 x.im))) (* y.im (atan2 x.im x.re))))
29.0ms
(/ (exp (* (log x.im) y.re)) (exp (* (atan2 x.im x.re) y.im)))

prune483.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes345.0ms

Accuracy

95.6% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 7.0b

bsearch1.8s

end0.0ms

sample2.1m

Algorithm
intervals
Results
2.0m29728×body10240exit
4.0s8000×body80valid