Average Error: 32.0 → 0.4
Time: 5.0s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
↓
\[-\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\tan^{-1}_* \frac{im}{re}}{\log base}\right)\right) \]
(FPCore (re im base)
 :precision binary64
 (/
  (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))
↓
(FPCore (re im base)
 :precision binary64
 (- (expm1 (log1p (/ (- (atan2 im re)) (log base))))))
double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}
↓
double code(double re, double im, double base) {
	return -expm1(log1p((-atan2(im, re) / log(base))));
}

public static double code(double re, double im, double base) {
	return ((Math.atan2(im, re) * Math.log(base)) - (Math.log(Math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}
↓
public static double code(double re, double im, double base) {
	return -Math.expm1(Math.log1p((-Math.atan2(im, re) / Math.log(base))));
}

def code(re, im, base):
	return ((math.atan2(im, re) * math.log(base)) - (math.log(math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))
↓
def code(re, im, base):
	return -math.expm1(math.log1p((-math.atan2(im, re) / math.log(base))))

function code(re, im, base)
	return Float64(Float64(Float64(atan(im, re) * log(base)) - Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end
↓
function code(re, im, base)
	return Float64(-expm1(log1p(Float64(Float64(-atan(im, re)) / log(base)))))
end

code[re_, im_, base_] := N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_, base_] := (-N[(Exp[N[Log[1 + N[((-N[ArcTan[im / re], $MachinePrecision]) / N[Log[base], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision])

\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
-\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\tan^{-1}_* \frac{im}{re}}{\log base}\right)\right)

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

    \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

  2. Simplified0.3

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

  3. Taylor expanded in base around inf 0.3

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022153 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))