Average Error: 31.8 → 0.2
Time: 3.8m
Precision: 64
Internal Precision: 128
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]

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 31.8

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

  2. Simplified6.9

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

  3. Taylor expanded around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019010 +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: 3.8m)Debug log

sample7.2s

Algorithm
intervals

simplify103.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
103.0ms
(* (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))))

prune38.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 2.3b

localize60.0ms

Local error

Found 4 expressions with local error:

32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
0.2b
(* (atan2 x.im x.re) y.re)
0.1b
(fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))
0.0b
(pow (exp y.im) (atan2 x.im x.re))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
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×associate-*l*
3×unpow-prod-down
3×pow-unpow
2×fma-udef
1×*-commutative
1×pow-exp
1×sin-sum
1×pow-pow
1×pow-to-exp
Counts
4 → 51
Calls
4 calls:
Slowest
3.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
2.0ms
(* (atan2 x.im x.re) y.re)
1.0ms
(pow (exp y.im) (atan2 x.im x.re))
0.0ms
(fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))

series395.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
179.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
156.0ms
(fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))
36.0ms
(pow (exp y.im) (atan2 x.im x.re))
24.0ms
(* (atan2 x.im x.re) y.re)

simplify341.0ms

Counts
23 → 63
Calls
23 calls:
Slowest
141.0ms
(+ (* y.im (atan2 x.im x.re)) (+ (* 1/2 (* (pow y.im 2) (pow (atan2 x.im x.re) 2))) 1))
63.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
60.0ms
(- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re))))
21.0ms
(- (* (atan2 x.im x.re) y.re) (* y.im (log (/ 1 x.re))))
20.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ 1 x.re)))))

prune1.1s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize39.0ms

Local error

Found 4 expressions with local error:

32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
0.3b
(* (pow y.im 2) (pow (atan2 x.im x.re) 2))
0.2b
(* (atan2 x.im x.re) y.re)
0.2b
(* y.im (atan2 x.im x.re))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
10×associate-*r*
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
7×associate-*l*
6×prod-exp
6×add-cbrt-cube
5×pow-to-exp
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×*-commutative
3×unpow-prod-down
2×unpow2
2×pow-exp
1×fma-udef
1×cbrt-unprod
1×pow-prod-down
1×sin-sum
Counts
4 → 65
Calls
4 calls:
Slowest
9.0ms
(* (pow y.im 2) (pow (atan2 x.im x.re) 2))
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))

series173.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
128.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
18.0ms
(* (atan2 x.im x.re) y.re)
14.0ms
(* y.im (atan2 x.im x.re))
13.0ms
(* (pow y.im 2) (pow (atan2 x.im x.re) 2))

simplify496.0ms

Counts
35 → 77
Calls
35 calls:
Slowest
164.0ms
(* (* (* (pow y.im 2) (pow y.im 2)) (pow y.im 2)) (* (* (pow (atan2 x.im x.re) 2) (pow (atan2 x.im x.re) 2)) (pow (atan2 x.im x.re) 2)))
78.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
32.0ms
(+ (log (pow y.im 2)) (* (log (atan2 x.im x.re)) 2))
30.0ms
(+ (log (pow y.im 2)) (* (log (atan2 x.im x.re)) 2))
23.0ms
(+ (log (pow y.im 2)) (log (pow (atan2 x.im x.re) 2)))

prune1.7s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize22.0ms

Local error

Found 4 expressions with local error:

32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
32.7b
(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))))

rewrite7.0ms

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
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)))
1.0ms
(cbrt (sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re))))

series537.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
207.0ms
(cbrt (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)))
112.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
102.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

simplify897.0ms

Counts
18 → 55
Calls
18 calls:
Slowest
318.0ms
(- (+ (* 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))))))))
114.0ms
(pow (sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re))))) 1/3)
110.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
86.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
79.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))

prune1.4s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.1b

localize41.0ms

Local error

Found 4 expressions with local error:

32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
32.7b
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

rewrite16.0ms

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
4.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (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)))
4.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (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)))

series528.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
149.0ms
(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)))
124.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))
124.0ms
(sin (fma y.im (log (hypot x.re x.im)) (* (atan2 x.im x.re) y.re)))

simplify646.0ms

Counts
16 → 52
Calls
16 calls:
Slowest
123.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
121.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
95.0ms
(sin (- (* (atan2 x.im x.re) y.re) (* y.im (log (/ -1 x.re)))))
79.0ms
(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)))))

prune1.5s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.1b

regimes445.0ms

Accuracy

0% (0.0b remaining)

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

bsearch8.0ms

end0.0ms

sample3.5m

Algorithm
intervals