math.log10 on complex, imaginary part

Average Error: 0.8 → 0.4

Time: 26.6s

Precision: binary64

Cost: 26432

Math

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

↓

\[\frac{4}{\frac{4}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log 10}^{2}}} \cdot \frac{1}{\log 10}} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (/ 4.0 (* (/ 4.0 (/ (atan2 im re) (pow (log 10.0) 2.0))) (/ 1.0 (log 10.0)))))

C

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

↓

double code(double re, double im) {
	return 4.0 / ((4.0 / (atan2(im, re) / pow(log(10.0), 2.0))) * (1.0 / log(10.0)));
}

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 = 4.0d0 / ((4.0d0 / (atan2(im, re) / (log(10.0d0) ** 2.0d0))) * (1.0d0 / log(10.0d0)))
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 4.0 / ((4.0 / (Math.atan2(im, re) / Math.pow(Math.log(10.0), 2.0))) * (1.0 / Math.log(10.0)));
}

Python

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

↓

def code(re, im):
	return 4.0 / ((4.0 / (math.atan2(im, re) / math.pow(math.log(10.0), 2.0))) * (1.0 / math.log(10.0)))

Julia

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

↓

function code(re, im)
	return Float64(4.0 / Float64(Float64(4.0 / Float64(atan(im, re) / (log(10.0) ^ 2.0))) * Float64(1.0 / log(10.0))))
end

MATLAB

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

↓

function tmp = code(re, im)
	tmp = 4.0 / ((4.0 / (atan2(im, re) / (log(10.0) ^ 2.0))) * (1.0 / log(10.0)));
end

Wolfram

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

↓

code[re_, im_] := N[(4.0 / N[(N[(4.0 / N[(N[ArcTan[im / re], $MachinePrecision] / N[Power[N[Log[10.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.8

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

2. Applied egg-rr 9.9

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

3. Taylor expanded in im around 0 0.9

   \[\leadsto \log 10 \cdot \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log 10}^{2}}} \]

4. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\frac{4}{\frac{4}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log 10}^{2}}} \cdot \frac{1}{\log 10}}} \]

5. Final simplification 0.4

   \[\leadsto \frac{4}{\frac{4}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log 10}^{2}}} \cdot \frac{1}{\log 10}} \]

Alternatives

Alternative 1
Error	0.4
Cost	26304

\[\frac{4}{\frac{\frac{4}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log 10}^{2}}}}{\log 10}} \]

Alternative 2
Error	0.8
Cost	26240

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

Alternative 3
Error	0.8
Cost	26176

\[\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\frac{1}{\log 10}}}{{\log 10}^{2}} \]

Alternative 4
Error	0.8
Cost	13056

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

Reproduce

herbie shell --seed 2023065 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))