Average Error: 31.3 → 0.0
Time: 4.1s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.75}\right)}\right)}^{\left(\frac{1}{{\log 10}^{0.25}}\right)}\right) \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (log
  (pow
   (pow (hypot re im) (pow (log 10.0) -0.75))
   (/ 1.0 (pow (log 10.0) 0.25)))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}
↓
double code(double re, double im) {
	return log(pow(pow(hypot(re, im), pow(log(10.0), -0.75)), (1.0 / pow(log(10.0), 0.25))));
}

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.pow(Math.pow(Math.hypot(re, im), Math.pow(Math.log(10.0), -0.75)), (1.0 / Math.pow(Math.log(10.0), 0.25))));
}

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.pow(math.pow(math.hypot(re, im), math.pow(math.log(10.0), -0.75)), (1.0 / math.pow(math.log(10.0), 0.25))))

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(((hypot(re, im) ^ (log(10.0) ^ -0.75)) ^ Float64(1.0 / (log(10.0) ^ 0.25))))
end

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end
↓
function tmp = code(re, im)
	tmp = log(((hypot(re, im) ^ (log(10.0) ^ -0.75)) ^ (1.0 / (log(10.0) ^ 0.25))));
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[Power[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Power[N[Log[10.0], $MachinePrecision], -0.75], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Power[N[Log[10.0], $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.75}\right)}\right)}^{\left(\frac{1}{{\log 10}^{0.25}}\right)}\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  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-rr0.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}{{\log 10}^{0.25}}}{{\log 10}^{0.25}}} \]

  5. Applied egg-rr0.4

    \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}{{\log 10}^{0.75}} \cdot \frac{1}{-{\log 10}^{0.25}}} \]

  6. Applied egg-rr0.0

    \[\leadsto \color{blue}{\log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.75}\right)}\right)}^{\left(\frac{1}{{\log 10}^{0.25}}\right)}\right)} \]

  7. Final simplification0.0

    \[\leadsto \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\log 10}^{-0.75}\right)}\right)}^{\left(\frac{1}{{\log 10}^{0.25}}\right)}\right) \]

Reproduce

herbie shell --seed 2022210 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))