Average Error: 32.6 → 0.2
Time: 15.7s
Precision: binary64
Cost: 38784
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}\right)}\right) \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (pow (sqrt (/ -1.0 (log 0.1))) 2.0))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}
↓
double code(double re, double im) {
	return log(pow(hypot(re, im), pow(sqrt((-1.0 / log(0.1))), 2.0)));
}

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.pow(Math.hypot(re, im), Math.pow(Math.sqrt((-1.0 / Math.log(0.1))), 2.0)));
}

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.pow(math.hypot(re, im), math.pow(math.sqrt((-1.0 / math.log(0.1))), 2.0)))

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((hypot(re, im) ^ (sqrt(Float64(-1.0 / log(0.1))) ^ 2.0)))
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) ^ (sqrt((-1.0 / log(0.1))) ^ 2.0)));
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[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Power[N[Sqrt[N[(-1.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}\right)}\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
    Proof
    (/.f64 (log.f64 (hypot.f64 re im)) (log.f64 10)): 0 points increase in error, 0 points decrease in error
    (/.f64 (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))))) (log.f64 10)): 118 points increase in error, 0 points decrease in error

  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-rr0.6

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

  5. Simplified0.3

    \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
    Proof
    (*.f64 (/.f64 (pow.f64 (log.f64 10) -1/2) (sqrt.f64 (log.f64 10))) (log.f64 (hypot.f64 re im))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 (log.f64 10) -1/2) (/.f64 (sqrt.f64 (log.f64 10)) (log.f64 (hypot.f64 re im))))): 90 points increase in error, 19 points decrease in error

  6. Applied egg-rr0.7

    \[\leadsto \color{blue}{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1}\right)}\right)} \]

  7. Applied egg-rr0.2

    \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}\right)}}\right) \]

  8. Final simplification0.2

    \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}\right)}\right) \]

Alternatives

Alternative 1
Error0.6
Cost19584
\[\frac{1}{\frac{\log 10}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
Alternative 2
Error0.6
Cost19456
\[\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]

Error

Reproduce

herbie shell --seed 2022334 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))