Average Error: 32.1 → 0.3
Time: 7.2s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
↓
\[\frac{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}{\log base} \]
(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))
↓
(FPCore (re im base)
 :precision binary64
 (/ (log (expm1 (log1p (hypot re im)))) (log base)))
double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}
↓
double code(double re, double im, double base) {
	return log(expm1(log1p(hypot(re, im)))) / log(base);
}

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

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

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

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $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[(N[Log[N[(Exp[N[Log[1 + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

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

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 32.1

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]

  3. Applied egg-rr0.3

    \[\leadsto \frac{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}}{\log base} \]

  4. Final simplification0.3

    \[\leadsto \frac{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}{\log base} \]

Reproduce

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