?

Average Error: 32.1 → 7.2
Time: 3.2s
Precision: binary64
Cost: 13512

?

\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -5.2 \cdot 10^{-145}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -5e+156)
   (log (- re))
   (if (<= re -5.2e-145) (log (sqrt (+ (* re re) (* im im)))) (log im))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -5e+156) {
		tmp = log(-re);
	} else if (re <= -5.2e-145) {
		tmp = log(sqrt(((re * re) + (im * im))));
	} else {
		tmp = log(im);
	}
	return tmp;
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im))))
end function
↓
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d+156)) then
        tmp = log(-re)
    else if (re <= (-5.2d-145)) then
        tmp = log(sqrt(((re * re) + (im * im))))
    else
        tmp = log(im)
    end if
    code = tmp
end function

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im))));
}
↓
public static double code(double re, double im) {
	double tmp;
	if (re <= -5e+156) {
		tmp = Math.log(-re);
	} else if (re <= -5.2e-145) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im))));
	} else {
		tmp = Math.log(im);
	}
	return tmp;
}

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im))))
↓
def code(re, im):
	tmp = 0
	if re <= -5e+156:
		tmp = math.log(-re)
	elif re <= -5.2e-145:
		tmp = math.log(math.sqrt(((re * re) + (im * im))))
	else:
		tmp = math.log(im)
	return tmp

function code(re, im)
	return log(sqrt(Float64(Float64(re * re) + Float64(im * im))))
end
↓
function code(re, im)
	tmp = 0.0
	if (re <= -5e+156)
		tmp = log(Float64(-re));
	elseif (re <= -5.2e-145)
		tmp = log(sqrt(Float64(Float64(re * re) + Float64(im * im))));
	else
		tmp = log(im);
	end
	return tmp
end

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im))));
end
↓
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e+156)
		tmp = log(-re);
	elseif (re <= -5.2e-145)
		tmp = log(sqrt(((re * re) + (im * im))));
	else
		tmp = log(im);
	end
	tmp_2 = tmp;
end

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]
↓
code[re_, im_] := If[LessEqual[re, -5e+156], N[Log[(-re)], $MachinePrecision], If[LessEqual[re, -5.2e-145], N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Log[im], $MachinePrecision]]]

\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -5 \cdot 10^{+156}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -5.2 \cdot 10^{-145}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log im\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 3 regimes

  2. if re < -4.99999999999999992e156

    1. Initial program 64.0

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

    2. Taylor expanded in re around -inf 3.4

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)} \]

    3. Simplified3.4

      \[\leadsto \log \color{blue}{\left(-re\right)} \]
      Proof

      [Start]3.4

      \[ \log \left(-1 \cdot re\right) \]

      rational_best-simplify-2 [=>]3.4

      \[ \log \color{blue}{\left(re \cdot -1\right)} \]

      rational_best-simplify-12 [=>]3.4

      \[ \log \color{blue}{\left(-re\right)} \]

    if -4.99999999999999992e156 < re < -5.1999999999999999e-145

    1. Initial program 11.1

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

    if -5.1999999999999999e-145 < re

    1. Initial program 32.8

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

    2. Taylor expanded in re around 0 5.7

      \[\leadsto \log \color{blue}{im} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -5.2 \cdot 10^{-145}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternatives

Alternative 1
Error10.1
Cost6924
\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -640000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.15 \cdot 10^{-23}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq -1.85 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]
Alternative 2
Error30.8
Cost6464
\[\log im \]

Error

Reproduce?

herbie shell --seed 2023092 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))