math.log10 on complex, imaginary part

Average Error: 0.9 → 0.2

Time: 5.2s

Precision: binary64

Cost: 13184

Math

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

↓

\[\frac{-1}{\frac{\log 0.1}{\tan^{-1}_* \frac{im}{re}}} \]

FPCore

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

↓

(FPCore (re im) :precision binary64 (/ -1.0 (/ (log 0.1) (atan2 im re))))

C

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

↓

double code(double re, double im) {
	return -1.0 / (log(0.1) / 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 = (-1.0d0) / (log(0.1d0) / 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 -1.0 / (Math.log(0.1) / Math.atan2(im, re));
}

Python

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

↓

def code(re, im):
	return -1.0 / (math.log(0.1) / math.atan2(im, re))

Julia

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

↓

function code(re, im)
	return Float64(-1.0 / Float64(log(0.1) / atan(im, re)))
end

MATLAB

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

↓

function tmp = code(re, im)
	tmp = -1.0 / (log(0.1) / atan2(im, re));
end

Wolfram

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

↓

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.9

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

2. Applied egg-rr 0.1

   \[\leadsto \color{blue}{-\frac{\tan^{-1}_* \frac{im}{re}}{\log 0.1}} \]

3. Applied egg-rr 0.2

   \[\leadsto -\color{blue}{{\left(\frac{\log 0.1}{\tan^{-1}_* \frac{im}{re}}\right)}^{-1}} \]

4. Applied egg-rr 0.2

   \[\leadsto -\color{blue}{\frac{1}{\frac{\log 0.1}{\tan^{-1}_* \frac{im}{re}}}} \]

5. Final simplification 0.2

   \[\leadsto \frac{-1}{\frac{\log 0.1}{\tan^{-1}_* \frac{im}{re}}} \]

Alternatives

Alternative 1
Error	0.1
Cost	13120

\[\frac{-\tan^{-1}_* \frac{im}{re}}{\log 0.1} \]

Alternative 2
Error	0.9
Cost	13056

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

Reproduce

herbie shell --seed 2022317 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))