Average Error: 32.2 → 0.2
Time: 5.0s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\right) \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (log
  (expm1
   (log1p
    (pow
     (pow (hypot re im) (sqrt (/ 1.0 (log 10.0))))
     (/ 1.0 (sqrt (log 10.0))))))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}
↓
double code(double re, double im) {
	return log(expm1(log1p(pow(pow(hypot(re, im), sqrt((1.0 / log(10.0)))), (1.0 / sqrt(log(10.0)))))));
}

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}
↓
public static double code(double re, double im) {
	return Math.log(Math.expm1(Math.log1p(Math.pow(Math.pow(Math.hypot(re, im), Math.sqrt((1.0 / Math.log(10.0)))), (1.0 / Math.sqrt(Math.log(10.0)))))));
}

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
↓
def code(re, im):
	return math.log(math.expm1(math.log1p(math.pow(math.pow(math.hypot(re, im), math.sqrt((1.0 / math.log(10.0)))), (1.0 / math.sqrt(math.log(10.0)))))))

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end
↓
function code(re, im)
	return log(expm1(log1p(((hypot(re, im) ^ sqrt(Float64(1.0 / log(10.0)))) ^ Float64(1.0 / sqrt(log(10.0)))))))
end

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_] := N[Log[N[(Exp[N[Log[1 + N[Power[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Sqrt[N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\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 32.2

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

  3. Applied add-sqr-sqrt_binary640.6

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}} \]

  4. Applied pow1_binary640.6

    \[\leadsto \frac{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \]

  5. Applied log-pow_binary640.6

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \]

  6. Applied times-frac_binary640.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]

  7. Applied add-log-exp_binary640.6

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\right)} \]

  8. Simplified0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)} \]

  9. Applied add-log-exp_binary640.3

    \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}\right)} \]

  10. Simplified0.1

    \[\leadsto \log \color{blue}{\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)} \]

  11. Applied expm1-log1p-u_binary640.2

    \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\right)} \]

  12. Final simplification0.2

    \[\leadsto \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))