Average Error: 31.7 → 0.2
Time: 3.7m
Precision: 64
Internal Precision: 128
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 31.7

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

  2. Simplified6.7

    \[\leadsto \color{blue}{\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}}\]
  3. Using strategy rm

  4. Applied pow-to-exp6.7

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{\color{blue}{e^{\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}}\]

  5. Applied pow-exp6.7

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}{e^{\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}\]

  6. Applied div-exp0.2

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}\]

  7. Final simplification0.2

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(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: 3.6m)Debug log

sample6.1s

Algorithm
intervals
Results
5.5s1377×body10240exit
71.0ms256×body80valid

simplify98.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
98.0ms
(* (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))))

prune34.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 3.1b

localize72.0ms

Local error

Found 4 expressions with local error:

36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
0.3b
(* (atan2 x.im x.re) y.re)
0.2b
(/ (pow (exp y.im) (atan2 x.im x.re)) (pow (hypot x.re x.im) y.re))
0.1b
(fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
42×unpow-prod-down
36×times-frac
33×add-cube-cbrt
33×*-un-lft-identity
33×add-sqr-sqrt
13×add-exp-log
9×div-exp
6×associate-/l*
6×pow-exp
6×add-cbrt-cube
6×associate-/r*
6×pow-to-exp
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-*l*
2×fma-udef
1×div-inv
1×*-commutative
1×sin-sum
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 103
Calls
4 calls:
Slowest
6.0ms
(/ (pow (exp y.im) (atan2 x.im x.re)) (pow (hypot x.re x.im) y.re))
4.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
3.0ms
(* (atan2 x.im x.re) y.re)
0.0ms
(fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))

series481.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
183.0ms
(/ (pow (exp y.im) (atan2 x.im x.re)) (pow (hypot x.re x.im) y.re))
140.0ms
(fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))
138.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
19.0ms
(* (atan2 x.im x.re) y.re)

simplify1.4s

Counts
108 → 115
Calls
108 calls:
Slowest
255.0ms
(/ (exp (* y.im (atan2 x.im x.re))) (exp (* -1 (* y.re (log (/ -1 x.im))))))
161.0ms
(/ (exp (* y.im (atan2 x.im x.re))) (exp (* -1 (* y.re (log (/ 1 x.im))))))
136.0ms
(/ (* (* (pow (exp y.im) (atan2 x.im x.re)) (pow (exp y.im) (atan2 x.im x.re))) (pow (exp y.im) (atan2 x.im x.re))) (* (* (pow (hypot x.re x.im) y.re) (pow (hypot x.re x.im) y.re)) (pow (hypot x.re x.im) y.re)))
76.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
71.0ms
(- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re))))

prune2.2s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.6b

localize26.0ms

Local error

Found 4 expressions with local error:

36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
0.3b
(* (log (hypot x.re x.im)) y.re)
0.3b
(* (atan2 x.im x.re) y.re)
0.2b
(* y.im (atan2 x.im x.re))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×associate-*l*
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×expm1-log1p-u
3×*-commutative
3×associate-*r*
1×fma-udef
1×log-pow
1×sin-sum
Counts
4 → 50
Calls
4 calls:
Slowest
6.0ms
(* (log (hypot x.re x.im)) y.re)
5.0ms
(* (atan2 x.im x.re) y.re)
4.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
2.0ms
(* y.im (atan2 x.im x.re))

series286.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
156.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
94.0ms
(* (log (hypot x.re x.im)) y.re)
22.0ms
(* (atan2 x.im x.re) y.re)
14.0ms
(* y.im (atan2 x.im x.re))

simplify344.0ms

Counts
20 → 62
Calls
20 calls:
Slowest
116.0ms
(* -1 (* y.re (log (/ -1 x.re))))
88.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
68.0ms
(* -1 (* y.re (log (/ 1 x.re))))
21.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ 1 x.re)))))
10.0ms
(+ (* (atan2 x.im x.re) y.re) (* y.im (log x.im)))

prune1.0s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.6b

localize28.0ms

Local error

Found 4 expressions with local error:

36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
0.6b
(cbrt (sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×fma-udef
3×sin-sum
3×cbrt-prod
1×pow1/3
Counts
4 → 43
Calls
4 calls:
Slowest
3.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
3.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
3.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
2.0ms
(cbrt (sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))))

series566.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
200.0ms
(cbrt (sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))))
133.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
117.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
116.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

simplify912.0ms

Counts
18 → 55
Calls
18 calls:
Slowest
373.0ms
(- (+ (* 1/3 (* y.im (* (exp (* 1/3 (- (log (/ 1 (pow (atan2 x.im x.re) 2))) (* 2 (log y.re))))) (log x.im)))) (exp (* 1/3 (+ (log (atan2 x.im x.re)) (log y.re))))) (* 7/81 (* (pow y.im 2) (* (pow (log x.im) 2) (exp (* 1/3 (+ (log (atan2 x.im x.re)) (log y.re))))))))
94.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
94.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
86.0ms
(pow (sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re))))) 1/3)
59.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))

prune1.1s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.5b

localize30.0ms

Local error

Found 4 expressions with local error:

36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
36.1b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×log1p-expm1-u
4×fma-udef
4×add-cube-cbrt
4×add-exp-log
4×sin-sum
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×expm1-log1p-u
4×add-sqr-sqrt
Counts
4 → 40
Calls
4 calls:
Slowest
2.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
2.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
2.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
2.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

series510.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
137.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
136.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
131.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
105.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

simplify580.0ms

Counts
16 → 52
Calls
16 calls:
Slowest
103.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
98.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
94.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
87.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
41.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ 1 x.re)))))

prune1.3s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.5b

regimes396.0ms

Accuracy

0% (0.0b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 0.2b

bsearch3.0ms

end0.0ms

sample3.3m

Algorithm
intervals
Results
3.2m46229×body10240exit
3.3s8000×body80valid