math.log10 on complex, imaginary part

Average Accuracy: 98.7% → 99.8%

Time: 5.6s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double re, double im) {
	return -atan2(im, re) / log(0.1);
}

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 = -atan2(im, re) / log(0.1d0)
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 -Math.atan2(im, re) / Math.log(0.1);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 98.7%

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

2. Applied egg-rr 99.8%

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

   [Start] 98.7	\[ \frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
   frac-2neg [=>] 98.7	\[ \color{blue}{\frac{-\tan^{-1}_* \frac{im}{re}}{-\log 10}} \]
   div-inv [=>] 98.6	\[ \color{blue}{\left(-\tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{-\log 10}} \]
   neg-log [=>] 99.8	\[ \left(-\tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]
   metadata-eval [=>] 99.8	\[ \left(-\tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\log \color{blue}{0.1}} \]

3. Simplified 99.8%

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

   [Start] 99.8	\[ \left(-\tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\log 0.1} \]
   associate-*r/ [=>] 99.8	\[ \color{blue}{\frac{\left(-\tan^{-1}_* \frac{im}{re}\right) \cdot 1}{\log 0.1}} \]
   *-rgt-identity [=>] 99.8	\[ \frac{\color{blue}{-\tan^{-1}_* \frac{im}{re}}}{\log 0.1} \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	13056

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

Reproduce

herbie shell --seed 2023151 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))