Average Error: 31.8 → 0.5
Time: 12.1s
Precision: binary64
Cost: 38848
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (/ (pow (log 10.0) -0.5) (/ (sqrt (log 10.0)) (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(log(10.0), -0.5) / (sqrt(log(10.0)) / 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.log(10.0), -0.5) / (Math.sqrt(Math.log(10.0)) / 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.log(10.0), -0.5) / (math.sqrt(math.log(10.0)) / 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((log(10.0) ^ -0.5) / Float64(sqrt(log(10.0)) / log(hypot(re, im))))
end

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end
↓
function tmp = code(re, im)
	tmp = (log(10.0) ^ -0.5) / (sqrt(log(10.0)) / 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[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision] / N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.8

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

  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)): 129 points increase in error, 0 points decrease in error

  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-rr0.5

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

  5. Final simplification0.5

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

Alternatives

Alternative 1
Error0.6
Cost19520
\[-\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1} \]
Alternative 2
Error0.6
Cost19456
\[\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]
Alternative 3
Error36.0
Cost13516
\[\begin{array}{l} t_0 := \frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{if}\;re \leq -5.342369402032849 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.785974809656684 \cdot 10^{-23}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \mathbf{elif}\;re \leq -2.1921059898248185 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]
Alternative 4
Error36.0
Cost13516
\[\begin{array}{l} t_0 := \frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{if}\;re \leq -5.342369402032849 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.785974809656684 \cdot 10^{-23}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \mathbf{elif}\;re \leq -2.1921059898248185 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]
Alternative 5
Error46.6
Cost12992
\[\frac{\log im}{\log 10} \]

Error

Reproduce

herbie shell --seed 2022291 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))