math.square on complex, real part

Percentage Accurate: 93.5% → 100.0%

Time: 8.3s

Alternatives: 3

Speedup: 1.2×

• 44.4% 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: 93.5% 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, 1.2× 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 96.1%

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

   1. difference-of-squares N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re - im \cdot im\\ t_1 := 0 - im \cdot im\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (let* ((t_0 (- (* re re) (* im im))) (t_1 (- 0.0 (* im im))))
   (if (<= t_0 -5e-315) t_1 (if (<= t_0 INFINITY) (* re re) t_1))))

C

double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double t_1 = 0.0 - (im * im);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = re * re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double t_1 = 0.0 - (im * im);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = re * re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	t_0 = (re * re) - (im * im)
	t_1 = 0.0 - (im * im)
	tmp = 0
	if t_0 <= -5e-315:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = re * re
	else:
		tmp = t_1
	return tmp

Julia

function re_sqr(re, im)
	t_0 = Float64(Float64(re * re) - Float64(im * im))
	t_1 = Float64(0.0 - Float64(im * im))
	tmp = 0.0
	if (t_0 <= -5e-315)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(re * re);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	t_0 = (re * re) - (im * im);
	t_1 = 0.0 - (im * im);
	tmp = 0.0;
	if (t_0 <= -5e-315)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = re * re;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(0.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-315], t$95$1, If[LessEqual[t$95$0, Infinity], N[(re * re), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -5.0000000023e-315 or +inf.0 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 92.7%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({im}^{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {im}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - {im}^{2}} \]

      4. +-rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left({im}^{2} + 0\right)} \]

      5. unpow2 N/A

         \[\leadsto 0 - \left(\color{blue}{im \cdot im} + 0\right) \]

      6. accelerator-lowering-fma.f64 98.5

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(im, im, 0\right)} \]

   5. Simplified 98.5%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(im, im, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 0 - \color{blue}{im \cdot im} \]

      2. *-lowering-*.f64 98.5

         \[\leadsto 0 - \color{blue}{im \cdot im} \]

   7. Applied egg-rr 98.5%

      \[\leadsto 0 - \color{blue}{im \cdot im} \]

   if -5.0000000023e-315 < (-.f64 (*.f64 re re) (*.f64 im im)) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{2}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{re}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{re \cdot re} + 0 \]

      3. accelerator-lowering-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 53.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in re around inf

   \[\leadsto \color{blue}{{re}^{2}} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{re}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{re \cdot re} + 0 \]

   3. accelerator-lowering-fma.f64 48.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]

5. Simplified 48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 48.3

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

7. Applied egg-rr 48.3%

   \[\leadsto \color{blue}{re \cdot re} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))