_divideComplex, real part

Percentage Accurate: 61.3% → 81.6%

Time: 9.6s

Alternatives: 7

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -1.5e+71)
     t_1
     (if (<= y.im -9.2e-151)
       t_0
       (if (<= y.im 6e-37)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 2.4e+71) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.5e+71) {
		tmp = t_1;
	} else if (y_46_im <= -9.2e-151) {
		tmp = t_0;
	} else if (y_46_im <= 6e-37) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 2.4e+71) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.5e+71)
		tmp = t_1;
	elseif (y_46_im <= -9.2e-151)
		tmp = t_0;
	elseif (y_46_im <= 6e-37)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 2.4e+71)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.5e+71], t$95$1, If[LessEqual[y$46$im, -9.2e-151], t$95$0, If[LessEqual[y$46$im, 6e-37], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.4e+71], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.50000000000000006e71 or 2.39999999999999981e71 < y.im

   1. Initial program 39.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im} + x.im}}{y.im} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot \frac{y.re}{y.im}} + x.im}{y.im} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}}{y.im} \]

      5. lower-/.f64 84.0

         \[\leadsto \frac{\mathsf{fma}\left(x.re, \color{blue}{\frac{y.re}{y.im}}, x.im\right)}{y.im} \]

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}} \]

   if -1.50000000000000006e71 < y.im < -9.19999999999999984e-151 or 6e-37 < y.im < 2.39999999999999981e71

   1. Initial program 82.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   if -9.19999999999999984e-151 < y.im < 6e-37

   1. Initial program 70.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re} + x.re}}{y.re} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{x.im \cdot \frac{y.im}{y.re}} + x.re}{y.re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}}{y.re} \]

      5. lower-/.f64 91.3

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{y.re}}, x.re\right)}{y.re} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+26}:\\ \;\;\;\;x.re \cdot \frac{y.re}{t\_0}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-154}:\\ \;\;\;\;\frac{y.im \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 11200:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.im -2.9e+69)
     (/ x.im y.im)
     (if (<= y.im -4.5e+26)
       (* x.re (/ y.re t_0))
       (if (<= y.im -1.15e-154)
         (/ (* y.im x.im) t_0)
         (if (<= y.im 11200.0) (/ x.re y.re) (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -2.9e+69) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4.5e+26) {
		tmp = x_46_re * (y_46_re / t_0);
	} else if (y_46_im <= -1.15e-154) {
		tmp = (y_46_im * x_46_im) / t_0;
	} else if (y_46_im <= 11200.0) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -2.9e+69)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -4.5e+26)
		tmp = Float64(x_46_re * Float64(y_46_re / t_0));
	elseif (y_46_im <= -1.15e-154)
		tmp = Float64(Float64(y_46_im * x_46_im) / t_0);
	elseif (y_46_im <= 11200.0)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.9e+69], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.5e+26], N[(x$46$re * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.15e-154], N[(N[(y$46$im * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 11200.0], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.8999999999999998e69 or 11200 < y.im

   1. Initial program 46.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 67.0

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -2.8999999999999998e69 < y.im < -4.49999999999999978e26

   1. Initial program 73.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 41.4

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   6. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2} + {y.re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x.re \cdot \frac{y.re}{{y.im}^{2} + {y.re}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot \frac{y.re}{{y.im}^{2} + {y.re}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x.re \cdot \color{blue}{\frac{y.re}{{y.im}^{2} + {y.re}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x.re \cdot \frac{y.re}{\color{blue}{y.im \cdot y.im} + {y.re}^{2}} \]

      5. lower-fma.f64 N/A

         \[\leadsto x.re \cdot \frac{y.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \]

      7. lower-*.f64 37.3

         \[\leadsto x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \]

   8. Applied rewrites 37.3%

      \[\leadsto \color{blue}{x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \]

   if -4.49999999999999978e26 < y.im < -1.15e-154

   1. Initial program 77.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x.im \cdot y.im}}{{y.im}^{2} + {y.re}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{x.im \cdot y.im}{\color{blue}{y.im \cdot y.im} + {y.re}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{x.im \cdot y.im}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{x.im \cdot y.im}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \]

      6. lower-*.f64 48.2

         \[\leadsto \frac{x.im \cdot y.im}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \]

   if -1.15e-154 < y.im < 11200

   1. Initial program 72.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 66.8

         \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+26}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-154}:\\ \;\;\;\;\frac{y.im \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 11200:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))