Average Error: 0.8 → 0.1
Time: 1.8s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
↓
\[-\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 0.1}\right)\right) \]
(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (- (log1p (expm1 (/ (atan2 im re) (log 0.1))))))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	return -log1p(expm1((atan2(im, re) / log(0.1))));
}

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.log1p(Math.expm1((Math.atan2(im, re) / Math.log(0.1))));
}

def code(re, im):
	return math.atan2(im, re) / math.log(10.0)
↓
def code(re, im):
	return -math.log1p(math.expm1((math.atan2(im, re) / math.log(0.1))))

function code(re, im)
	return Float64(atan(im, re) / log(10.0))
end
↓
function code(re, im)
	return Float64(-log1p(expm1(Float64(atan(im, re) / log(0.1)))))
end

code[re_, im_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_] := (-N[Log[1 + N[(Exp[N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision])

\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
-\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 0.1}\right)\right)

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.8

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

  2. Applied egg-rr0.1

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

  3. Applied egg-rr0.3

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

  4. Applied egg-rr0.1

    \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 0.1}\right)\right)} \]

  5. Final simplification0.1

    \[\leadsto -\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 0.1}\right)\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))