math.square on complex, real part

Percentage Accurate: 93.7% → 100.0%

Time: 4.5s

Alternatives: 4

Speedup: 0.6×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 10^{+142}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(1 - \frac{im}{re} \cdot \frac{im}{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 1e+142)
   (- (* re re) (* im im))
   (* (pow re 2.0) (- 1.0 (* (/ im re) (/ im re))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 1e+142) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = pow(re, 2.0) * (1.0 - ((im / re) * (im / re)));
	}
	return tmp;
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re * re) <= 1d+142) then
        tmp = (re * re) - (im * im)
    else
        tmp = (re ** 2.0d0) * (1.0d0 - ((im / re) * (im / re)))
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 1e+142) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = Math.pow(re, 2.0) * (1.0 - ((im / re) * (im / re)));
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 1e+142:
		tmp = (re * re) - (im * im)
	else:
		tmp = math.pow(re, 2.0) * (1.0 - ((im / re) * (im / re)))
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 1e+142)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64((re ^ 2.0) * Float64(1.0 - Float64(Float64(im / re) * Float64(im / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 1e+142)
		tmp = (re * re) - (im * im);
	else
		tmp = (re ^ 2.0) * (1.0 - ((im / re) * (im / re)));
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 1e+142], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[Power[re, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 1.00000000000000005e142

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   if 1.00000000000000005e142 < (*.f64 re re)

   1. Initial program 84.9%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 84.9%

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(1 + -1 \cdot \frac{{im}^{2}}{{re}^{2}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 84.9%

         \[\leadsto {re}^{2} \cdot \left(1 + \color{blue}{\left(-\frac{{im}^{2}}{{re}^{2}}\right)}\right) \]

      2. unsub-neg 84.9%

         \[\leadsto {re}^{2} \cdot \color{blue}{\left(1 - \frac{{im}^{2}}{{re}^{2}}\right)} \]

   5. Simplified 84.9%

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(1 - \frac{{im}^{2}}{{re}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 84.9%

         \[\leadsto {re}^{2} \cdot \left(1 - \frac{{im}^{2}}{\color{blue}{re \cdot re}}\right) \]

      2. unpow2 84.9%

         \[\leadsto {re}^{2} \cdot \left(1 - \frac{\color{blue}{im \cdot im}}{re \cdot re}\right) \]

      3. times-frac 100.0%

         \[\leadsto {re}^{2} \cdot \left(1 - \color{blue}{\frac{im}{re} \cdot \frac{im}{re}}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {re}^{2} \cdot \left(1 - \color{blue}{\frac{im}{re} \cdot \frac{im}{re}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 1e+198) (fma re re (* im (- im))) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 1e+198) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 1e+198)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(Float64(re + im) * Float64(re + im));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, 1e+198], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.00000000000000002e198

   1. Initial program 96.1%

      \[re \cdot re - im \cdot im \]
   2. Step-by-step derivation

      1. sqr-neg 96.1%

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

      2. cancel-sign-sub 96.1%

         \[\leadsto \color{blue}{re \cdot re + im \cdot \left(-im\right)} \]

      3. fma-define 98.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)} \]
   4. Add Preprocessing

   if 1.00000000000000002e198 < re

   1. Initial program 78.3%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. difference-of-squares 100.0%

         \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re - im\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \left(re + im\right) \cdot \color{blue}{\left(re + \left(-im\right)\right)} \]

      3. add-sqr-sqrt 60.9%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}\right) \]

      4. sqrt-unprod 95.7%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}\right) \]

      5. sqr-neg 95.7%

         \[\leadsto \left(re + im\right) \cdot \left(re + \sqrt{\color{blue}{im \cdot im}}\right) \]

      6. sqrt-prod 34.8%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{im} \cdot \sqrt{im}}\right) \]

      7. add-sqr-sqrt 95.7%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{im}\right) \]

   4. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re + im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 5.8e+150) (- (* re re) (* im im)) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 5.8e+150) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+150) then
        tmp = (re * re) - (im * im)
    else
        tmp = (re + im) * (re + im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if (re <= 5.8e+150) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if re <= 5.8e+150:
		tmp = (re * re) - (im * im)
	else:
		tmp = (re + im) * (re + im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 5.8e+150)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(Float64(re + im) * Float64(re + im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (re <= 5.8e+150)
		tmp = (re * re) - (im * im);
	else
		tmp = (re + im) * (re + im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, 5.8e+150], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.80000000000000022e150

   1. Initial program 97.3%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   if 5.80000000000000022e150 < re

   1. Initial program 75.8%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. difference-of-squares 100.0%

         \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re - im\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \left(re + im\right) \cdot \color{blue}{\left(re + \left(-im\right)\right)} \]

      3. add-sqr-sqrt 57.6%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}\right) \]

      4. sqrt-unprod 90.9%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}\right) \]

      5. sqr-neg 90.9%

         \[\leadsto \left(re + im\right) \cdot \left(re + \sqrt{\color{blue}{im \cdot im}}\right) \]

      6. sqrt-prod 33.3%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{im} \cdot \sqrt{im}}\right) \]

      7. add-sqr-sqrt 87.9%

         \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{im}\right) \]

   4. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re + im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(re + im\right) \cdot \left(re + im\right) \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (* (+ re im) (+ re im)))

C

double re_sqr(double re, double im) {
	return (re + im) * (re + im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re + im) * (re + im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re + im) * (re + im);
}

Python

def re_sqr(re, im):
	return (re + im) * (re + im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re + im) * Float64(re + im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re + im) * (re + im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-squares 100.0%

      \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re - im\right)} \]

   2. sub-neg 100.0%

      \[\leadsto \left(re + im\right) \cdot \color{blue}{\left(re + \left(-im\right)\right)} \]

   3. add-sqr-sqrt 51.9%

      \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}\right) \]

   4. sqrt-unprod 74.5%

      \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}\right) \]

   5. sqr-neg 74.5%

      \[\leadsto \left(re + im\right) \cdot \left(re + \sqrt{\color{blue}{im \cdot im}}\right) \]

   6. sqrt-prod 24.1%

      \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{\sqrt{im} \cdot \sqrt{im}}\right) \]

   7. add-sqr-sqrt 52.8%

      \[\leadsto \left(re + im\right) \cdot \left(re + \color{blue}{im}\right) \]

4. Applied egg-rr 52.8%

   \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re + im\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))