?

Average Error: 31.6 → 7.5
Time: 6.2s
Precision: binary64
Cost: 7112

?

\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]
\[\sqrt{re \cdot re + im \cdot im} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+90}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore modulus (re im)
 :precision binary64
 (if (<= re -2.8e+90)
   (- re)
   (if (<= re -7.2e-130) (sqrt (+ (* re re) (* im im))) im)))
double modulus(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}
↓
double modulus(double re, double im) {
	double tmp;
	if (re <= -2.8e+90) {
		tmp = -re;
	} else if (re <= -7.2e-130) {
		tmp = sqrt(((re * re) + (im * im)));
	} else {
		tmp = im;
	}
	return tmp;
}

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus = sqrt(((re * re) + (im * im)))
end function
↓
real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.8d+90)) then
        tmp = -re
    else if (re <= (-7.2d-130)) then
        tmp = sqrt(((re * re) + (im * im)))
    else
        tmp = im
    end if
    modulus = tmp
end function

public static double modulus(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}
↓
public static double modulus(double re, double im) {
	double tmp;
	if (re <= -2.8e+90) {
		tmp = -re;
	} else if (re <= -7.2e-130) {
		tmp = Math.sqrt(((re * re) + (im * im)));
	} else {
		tmp = im;
	}
	return tmp;
}

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))
↓
def modulus(re, im):
	tmp = 0
	if re <= -2.8e+90:
		tmp = -re
	elif re <= -7.2e-130:
		tmp = math.sqrt(((re * re) + (im * im)))
	else:
		tmp = im
	return tmp

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end
↓
function modulus(re, im)
	tmp = 0.0
	if (re <= -2.8e+90)
		tmp = Float64(-re);
	elseif (re <= -7.2e-130)
		tmp = sqrt(Float64(Float64(re * re) + Float64(im * im)));
	else
		tmp = im;
	end
	return tmp
end

function tmp = modulus(re, im)
	tmp = sqrt(((re * re) + (im * im)));
end
↓
function tmp_2 = modulus(re, im)
	tmp = 0.0;
	if (re <= -2.8e+90)
		tmp = -re;
	elseif (re <= -7.2e-130)
		tmp = sqrt(((re * re) + (im * im)));
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

modulus[re_, im_] := N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
modulus[re_, im_] := If[LessEqual[re, -2.8e+90], (-re), If[LessEqual[re, -7.2e-130], N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], im]]

\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.8 \cdot 10^{+90}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -7.2 \cdot 10^{-130}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;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 < -2.8e90

    1. Initial program 49.9

      \[\sqrt{re \cdot re + im \cdot im} \]

    2. Taylor expanded in re around -inf 6.1

      \[\leadsto \color{blue}{-1 \cdot re} \]

    3. Simplified6.1

      \[\leadsto \color{blue}{-re} \]
      Proof

      [Start]6.1

      \[ -1 \cdot re \]

      rational_best-simplify-1 [=>]6.1

      \[ \color{blue}{re \cdot -1} \]

      rational_best-simplify-10 [=>]6.1

      \[ \color{blue}{-re} \]

    if -2.8e90 < re < -7.2000000000000003e-130

    1. Initial program 10.7

      \[\sqrt{re \cdot re + im \cdot im} \]

    if -7.2000000000000003e-130 < re

    1. Initial program 30.5

      \[\sqrt{re \cdot re + im \cdot im} \]

    2. Taylor expanded in re around 0 6.5

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+90}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternatives

Alternative 1
Error10.6
Cost260
\[\begin{array}{l} \mathbf{if}\;re \leq -5.4 \cdot 10^{-127}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
Alternative 2
Error31.1
Cost64
\[im \]

Error

Reproduce?

herbie shell --seed 2023099 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))