math.log10 on complex, imaginary part

Average Error: 0.8 → 0.3

Time: 9.6s

Precision: binary64

Cost: 20224

Math

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

↓

\[\frac{-0.5}{\log 10} \cdot \left(\left(0 - \frac{1}{\frac{0.3333333333333333}{\tan^{-1}_* \frac{im}{re}}}\right) - \left(-\tan^{-1}_* \frac{im}{re}\right)\right) \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))

↓

(FPCore (re im)
 :precision binary64
 (*
  (/ -0.5 (log 10.0))
  (- (- 0.0 (/ 1.0 (/ 0.3333333333333333 (atan2 im re)))) (- (atan2 im re)))))

C

double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}

↓

double code(double re, double im) {
	return (-0.5 / log(10.0)) * ((0.0 - (1.0 / (0.3333333333333333 / atan2(im, re)))) - -atan2(im, re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / log(10.0d0)
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-0.5d0) / log(10.0d0)) * ((0.0d0 - (1.0d0 / (0.3333333333333333d0 / atan2(im, re)))) - -atan2(im, re))
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / Math.log(10.0);
}

↓

public static double code(double re, double im) {
	return (-0.5 / Math.log(10.0)) * ((0.0 - (1.0 / (0.3333333333333333 / Math.atan2(im, re)))) - -Math.atan2(im, re));
}

Python

def code(re, im):
	return math.atan2(im, re) / math.log(10.0)

↓

def code(re, im):
	return (-0.5 / math.log(10.0)) * ((0.0 - (1.0 / (0.3333333333333333 / math.atan2(im, re)))) - -math.atan2(im, re))

Julia

function code(re, im)
	return Float64(atan(im, re) / log(10.0))
end

↓

function code(re, im)
	return Float64(Float64(-0.5 / log(10.0)) * Float64(Float64(0.0 - Float64(1.0 / Float64(0.3333333333333333 / atan(im, re)))) - Float64(-atan(im, re))))
end

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / log(10.0);
end

↓

function tmp = code(re, im)
	tmp = (-0.5 / log(10.0)) * ((0.0 - (1.0 / (0.3333333333333333 / atan2(im, re)))) - -atan2(im, re));
end

Wolfram

code[re_, im_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[(-0.5 / N[Log[10.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.0 - N[(1.0 / N[(0.3333333333333333 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-N[ArcTan[im / re], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.8

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

2. Applied egg-rr 0.9

   \[\leadsto \color{blue}{\frac{-0.5}{\log 10} \cdot \left(\left(0 - \tan^{-1}_* \frac{im}{re} \cdot 3\right) - \left(-\tan^{-1}_* \frac{im}{re}\right)\right)} \]

3. Applied egg-rr 0.3

   \[\leadsto \frac{-0.5}{\log 10} \cdot \left(\left(0 - \color{blue}{\frac{1}{\frac{0.3333333333333333}{\tan^{-1}_* \frac{im}{re}}}}\right) - \left(-\tan^{-1}_* \frac{im}{re}\right)\right) \]

4. Final simplification 0.3

   \[\leadsto \frac{-0.5}{\log 10} \cdot \left(\left(0 - \frac{1}{\frac{0.3333333333333333}{\tan^{-1}_* \frac{im}{re}}}\right) - \left(-\tan^{-1}_* \frac{im}{re}\right)\right) \]

Alternatives

Alternative 1
Error	0.8
Cost	19840

\[\tan^{-1}_* \frac{im}{re} \cdot \left(\frac{-0.5}{\log 10} - \frac{-1.5}{\log 10}\right) \]

Alternative 2
Error	0.8
Cost	13056

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

Reproduce

herbie shell --seed 2023066 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))