math.square on complex, real part

Percentage Accurate: 93.8% → 99.0%

Time: 4.5s

Alternatives: 4

Speedup: 0.6×

• 44.0% 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: 93.8% 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: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re - im \cdot im\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(1 - \frac{\frac{im}{re}}{\frac{re}{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (let* ((t_0 (- (* re re) (* im im))))
   (if (<= t_0 -5e-261) t_0 (* (pow re 2.0) (- 1.0 (/ (/ im re) (/ re im)))))))

C

double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double tmp;
	if (t_0 <= -5e-261) {
		tmp = t_0;
	} else {
		tmp = pow(re, 2.0) * (1.0 - ((im / re) / (re / im)));
	}
	return tmp;
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) - (im * im)
    if (t_0 <= (-5d-261)) then
        tmp = t_0
    else
        tmp = (re ** 2.0d0) * (1.0d0 - ((im / re) / (re / im)))
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double tmp;
	if (t_0 <= -5e-261) {
		tmp = t_0;
	} else {
		tmp = Math.pow(re, 2.0) * (1.0 - ((im / re) / (re / im)));
	}
	return tmp;
}

Python

def re_sqr(re, im):
	t_0 = (re * re) - (im * im)
	tmp = 0
	if t_0 <= -5e-261:
		tmp = t_0
	else:
		tmp = math.pow(re, 2.0) * (1.0 - ((im / re) / (re / im)))
	return tmp

Julia

function re_sqr(re, im)
	t_0 = Float64(Float64(re * re) - Float64(im * im))
	tmp = 0.0
	if (t_0 <= -5e-261)
		tmp = t_0;
	else
		tmp = Float64((re ^ 2.0) * Float64(1.0 - Float64(Float64(im / re) / Float64(re / im))));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	t_0 = (re * re) - (im * im);
	tmp = 0.0;
	if (t_0 <= -5e-261)
		tmp = t_0;
	else
		tmp = (re ^ 2.0) * (1.0 - ((im / re) / (re / im)));
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-261], t$95$0, N[(N[Power[re, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(im / re), $MachinePrecision] / N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -4.99999999999999981e-261

   1. Initial program 100.0%

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

   if -4.99999999999999981e-261 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 90.9%

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

   3. Taylor expanded in re around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

   5. Simplified 82.5%

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

      1. unpow2 82.5%

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

      2. associate-/l* 87.7%

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

   7. Applied egg-rr 87.7%

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

      1. *-un-lft-identity 87.7%

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

      2. unpow2 87.7%

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

      3. times-frac 98.1%

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

   9. Applied egg-rr 98.1%

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

       1. associate-*r* 100.0%

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

       2. div-inv 100.0%

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

       3. clear-num 100.0%

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

       4. un-div-inv 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq -5 \cdot 10^{-261}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(1 - \frac{\frac{im}{re}}{\frac{re}{im}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 5e+235) (fma re re (* im (- im))) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 5e+235) {
		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 <= 5e+235)
		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, 5e+235], 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 < 5.00000000000000027e235

   1. Initial program 95.8%

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

      1. sqr-neg 95.8%

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

      2. cancel-sign-sub 95.8%

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

      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 5.00000000000000027e235 < re

   1. Initial program 76.5%

      \[re \cdot re - im \cdot im \]
   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 52.9%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-prod 47.1%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 1.35e+154) (- (* re re) (* im im)) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 1.35e+154) {
		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 <= 1.35d+154) 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 <= 1.35e+154) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if re <= 1.35e+154:
		tmp = (re * re) - (im * im)
	else:
		tmp = (re + im) * (re + im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 1.35e+154)
		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 <= 1.35e+154)
		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, 1.35e+154], 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 < 1.35000000000000003e154

   1. Initial program 98.1%

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

   if 1.35000000000000003e154 < re

   1. Initial program 74.4%

      \[re \cdot re - im \cdot im \]
   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 53.8%

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

      4. sqrt-unprod 97.4%

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

      5. sqr-neg 97.4%

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

      6. sqrt-prod 43.6%

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

      7. add-sqr-sqrt 84.6%

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

   4. Applied egg-rr 84.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.4% 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 53.0%

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

   4. sqrt-unprod 78.7%

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

   5. sqr-neg 78.7%

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

   6. sqrt-prod 27.2%

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

   7. add-sqr-sqrt 57.2%

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

4. Applied egg-rr 57.2%

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

5. Final simplification 57.2%

   \[\leadsto \left(re + im\right) \cdot \left(re + im\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))