Average Error: 31.7 → 0.3
Time: 2.9s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{-\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (* (/ (pow (log1p 9.0) -0.5) (- (sqrt (log1p 9.0)))) (- (log (hypot re im)))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}
↓
double code(double re, double im) {
	return (pow(log1p(9.0), -0.5) / -sqrt(log1p(9.0))) * -log(hypot(re, im));
}

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.pow(Math.log1p(9.0), -0.5) / -Math.sqrt(Math.log1p(9.0))) * -Math.log(Math.hypot(re, im));
}

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
↓
def code(re, im):
	return (math.pow(math.log1p(9.0), -0.5) / -math.sqrt(math.log1p(9.0))) * -math.log(math.hypot(re, im))

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end
↓
function code(re, im)
	return Float64(Float64((log1p(9.0) ^ -0.5) / Float64(-sqrt(log1p(9.0)))) * Float64(-log(hypot(re, im))))
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[(N[(N[Power[N[Log[1 + 9.0], $MachinePrecision], -0.5], $MachinePrecision] / (-N[Sqrt[N[Log[1 + 9.0], $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * (-N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{-\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\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 31.7

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

  2. Simplified0.6

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

  3. Applied egg-rr0.5

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

  4. Applied egg-rr0.6

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

  5. Applied egg-rr0.3

    \[\leadsto \color{blue}{\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{-\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \]

  6. Final simplification0.3

    \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{-\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))