Average Error: 32.4 → 0.2
Time: 4.6s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\\ t_1 := {\log 10}^{-0.5}\\ \log \left({\left({\left({t_0}^{2}\right)}^{t_1} \cdot {t_0}^{t_1}\right)}^{t_1}\right) \end{array} \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (cbrt (hypot re im))) (t_1 (pow (log 10.0) -0.5)))
   (log (pow (* (pow (pow t_0 2.0) t_1) (pow t_0 t_1)) t_1))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}
↓
double code(double re, double im) {
	double t_0 = cbrt(hypot(re, im));
	double t_1 = pow(log(10.0), -0.5);
	return log(pow((pow(pow(t_0, 2.0), t_1) * pow(t_0, t_1)), t_1));
}

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) {
	double t_0 = Math.cbrt(Math.hypot(re, im));
	double t_1 = Math.pow(Math.log(10.0), -0.5);
	return Math.log(Math.pow((Math.pow(Math.pow(t_0, 2.0), t_1) * Math.pow(t_0, t_1)), t_1));
}

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end
↓
function code(re, im)
	t_0 = cbrt(hypot(re, im))
	t_1 = log(10.0) ^ -0.5
	return log((Float64(((t_0 ^ 2.0) ^ t_1) * (t_0 ^ t_1)) ^ t_1))
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_] := Block[{t$95$0 = N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision]}, N[Log[N[Power[N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], t$95$1], $MachinePrecision] * N[Power[t$95$0, t$95$1], $MachinePrecision]), $MachinePrecision], t$95$1], $MachinePrecision]], $MachinePrecision]]]

\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\\
t_1 := {\log 10}^{-0.5}\\
\log \left({\left({\left({t_0}^{2}\right)}^{t_1} \cdot {t_0}^{t_1}\right)}^{t_1}\right)
\end{array}

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 32.4

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

  2. Simplified0.6

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

  3. Applied egg-rr0.6

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

  4. Applied egg-rr0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.5}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.5}\right)}}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.5}\right)}}\right)\right)} \]

  5. Applied egg-rr0.2

    \[\leadsto \color{blue}{\log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.5}\right)}\right)}^{\left({\log 10}^{-0.5}\right)}\right)} \]

  6. Applied egg-rr0.2

    \[\leadsto \log \left({\color{blue}{\left({\left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right)}^{\left({\log 10}^{-0.5}\right)} \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{\left({\log 10}^{-0.5}\right)}\right)}}^{\left({\log 10}^{-0.5}\right)}\right) \]

  7. Final simplification0.2

    \[\leadsto \log \left({\left({\left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right)}^{\left({\log 10}^{-0.5}\right)} \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{\left({\log 10}^{-0.5}\right)}\right)}^{\left({\log 10}^{-0.5}\right)}\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))