_divideComplex, real part

Percentage Accurate: 61.7% → 82.9%

Time: 3.5s

Alternatives: 11

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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: 61.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -7.4 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-150}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(y.re, x.re, -\frac{\left(y.re \cdot y.re\right) \cdot x.im}{y.im}\right)}{y.im}, -1, -x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -4e+128)
     t_1
     (if (<= y.re -7.4e-119)
       t_0
       (if (<= y.re 1.02e-150)
         (-
          (/
           (fma
            (/ (fma y.re x.re (- (/ (* (* y.re y.re) x.im) y.im))) y.im)
            -1.0
            (- x.im))
           y.im))
         (if (<= y.re 3e+89) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -4e+128) {
		tmp = t_1;
	} else if (y_46_re <= -7.4e-119) {
		tmp = t_0;
	} else if (y_46_re <= 1.02e-150) {
		tmp = -(fma((fma(y_46_re, x_46_re, -(((y_46_re * y_46_re) * x_46_im) / y_46_im)) / y_46_im), -1.0, -x_46_im) / y_46_im);
	} else if (y_46_re <= 3e+89) {
		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(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -4e+128)
		tmp = t_1;
	elseif (y_46_re <= -7.4e-119)
		tmp = t_0;
	elseif (y_46_re <= 1.02e-150)
		tmp = Float64(-Float64(fma(Float64(fma(y_46_re, x_46_re, Float64(-Float64(Float64(Float64(y_46_re * y_46_re) * x_46_im) / y_46_im))) / y_46_im), -1.0, Float64(-x_46_im)) / y_46_im));
	elseif (y_46_re <= 3e+89)
		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[(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]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -4e+128], t$95$1, If[LessEqual[y$46$re, -7.4e-119], t$95$0, If[LessEqual[y$46$re, 1.02e-150], (-N[(N[(N[(N[(y$46$re * x$46$re + (-N[(N[(N[(y$46$re * y$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision])), $MachinePrecision] / y$46$im), $MachinePrecision] * -1.0 + (-x$46$im)), $MachinePrecision] / y$46$im), $MachinePrecision]), If[LessEqual[y$46$re, 3e+89], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.0000000000000003e128 or 3.00000000000000013e89 < y.re

   1. Initial program 38.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.1

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

   4. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 84.3

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

   6. Applied rewrites 84.3%

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

   if -4.0000000000000003e128 < y.re < -7.4000000000000003e-119 or 1.0199999999999999e-150 < y.re < 3.00000000000000013e89

   1. Initial program 76.1%

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

   if -7.4000000000000003e-119 < y.re < 1.0199999999999999e-150

   1. Initial program 69.7%

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

   2. Taylor expanded in y.im around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot x.im + -1 \cdot \frac{-1 \cdot \frac{x.im \cdot {y.re}^{2}}{y.im} + x.re \cdot y.re}{y.im}}{y.im}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot x.im + -1 \cdot \frac{-1 \cdot \frac{x.im \cdot {y.re}^{2}}{y.im} + x.re \cdot y.re}{y.im}}{y.im}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot x.im + -1 \cdot \frac{-1 \cdot \frac{x.im \cdot {y.re}^{2}}{y.im} + x.re \cdot y.re}{y.im}}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot x.im + -1 \cdot \frac{-1 \cdot \frac{x.im \cdot {y.re}^{2}}{y.im} + x.re \cdot y.re}{y.im}}{y.im} \]

   4. Applied rewrites 91.0%

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

Alternative 2: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -7.4 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.re + x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -4e+128)
     t_1
     (if (<= y.re -7.4e-119)
       t_0
       (if (<= y.re 1.02e-150)
         (/ (+ (* (/ y.re y.im) x.re) x.im) y.im)
         (if (<= y.re 3e+89) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -4e+128) {
		tmp = t_1;
	} else if (y_46_re <= -7.4e-119) {
		tmp = t_0;
	} else if (y_46_re <= 1.02e-150) {
		tmp = (((y_46_re / y_46_im) * x_46_re) + x_46_im) / y_46_im;
	} else if (y_46_re <= 3e+89) {
		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(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -4e+128)
		tmp = t_1;
	elseif (y_46_re <= -7.4e-119)
		tmp = t_0;
	elseif (y_46_re <= 1.02e-150)
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_re) + x_46_im) / y_46_im);
	elseif (y_46_re <= 3e+89)
		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[(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]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -4e+128], t$95$1, If[LessEqual[y$46$re, -7.4e-119], t$95$0, If[LessEqual[y$46$re, 1.02e-150], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3e+89], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.0000000000000003e128 or 3.00000000000000013e89 < y.re

   1. Initial program 38.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.1

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

   4. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 84.3

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

   6. Applied rewrites 84.3%

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

   if -4.0000000000000003e128 < y.re < -7.4000000000000003e-119 or 1.0199999999999999e-150 < y.re < 3.00000000000000013e89

   1. Initial program 76.1%

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

   if -7.4000000000000003e-119 < y.re < 1.0199999999999999e-150

   1. Initial program 69.7%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 91.4

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

   6. Applied rewrites 91.4%

      \[\leadsto \frac{\frac{y.re}{y.im} \cdot x.re + x.im}{y.im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)\\ \mathbf{if}\;y.im \leq -4000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+76}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im x.im (* y.re x.re))))
   (if (<= y.im -4000000.0)
     (/ x.im y.im)
     (if (<= y.im 5.3e-42)
       (/ t_0 (* y.re y.re))
       (if (<= y.im 9e+76) (/ t_0 (* y.im y.im)) (/ 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, x_46_im, (y_46_re * x_46_re));
	double tmp;
	if (y_46_im <= -4000000.0) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.3e-42) {
		tmp = t_0 / (y_46_re * y_46_re);
	} else if (y_46_im <= 9e+76) {
		tmp = t_0 / (y_46_im * y_46_im);
	} 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, x_46_im, Float64(y_46_re * x_46_re))
	tmp = 0.0
	if (y_46_im <= -4000000.0)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 5.3e-42)
		tmp = Float64(t_0 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 9e+76)
		tmp = Float64(t_0 / Float64(y_46_im * y_46_im));
	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 * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4000000.0], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.3e-42], N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9e+76], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4e6 or 8.9999999999999995e76 < y.im

   1. Initial program 46.9%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 68.6

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

   4. Applied rewrites 68.6%

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

   if -4e6 < y.im < 5.3e-42

   1. Initial program 73.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in y.re around 0

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

      1. div-add N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 60.1

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

   7. Applied rewrites 60.1%

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

   if 5.3e-42 < y.im < 8.9999999999999995e76

   1. Initial program 74.7%

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

   2. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.4

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

   6. Applied rewrites 50.4%

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

Alternative 4: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7700000000:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.re + x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -2800000000.0)
     t_0
     (if (<= y.re 7700000000.0)
       (/ (+ (* (/ y.re y.im) x.re) x.im) y.im)
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -2800000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 7700000000.0) {
		tmp = (((y_46_re / y_46_im) * x_46_re) + x_46_im) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2800000000.0)
		tmp = t_0;
	elseif (y_46_re <= 7700000000.0)
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_re) + x_46_im) / y_46_im);
	else
		tmp = t_0;
	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$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2800000000.0], t$95$0, If[LessEqual[y$46$re, 7700000000.0], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.8e9 or 7.7e9 < y.re

   1. Initial program 48.7%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 75.9

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

   6. Applied rewrites 75.9%

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

   if -2.8e9 < y.re < 7.7e9

   1. Initial program 74.1%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.3

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

   4. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 79.3

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

   6. Applied rewrites 79.3%

      \[\leadsto \frac{\frac{y.re}{y.im} \cdot x.re + x.im}{y.im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7700000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -2800000000.0)
     t_0
     (if (<= y.re 7700000000.0) (/ (fma x.re (/ y.re y.im) x.im) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -2800000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 7700000000.0) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2800000000.0)
		tmp = t_0;
	elseif (y_46_re <= 7700000000.0)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = t_0;
	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$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2800000000.0], t$95$0, If[LessEqual[y$46$re, 7700000000.0], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.8e9 or 7.7e9 < y.re

   1. Initial program 48.7%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 75.9

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

   6. Applied rewrites 75.9%

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

   if -2.8e9 < y.re < 7.7e9

   1. Initial program 74.1%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.3

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

   4. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 145000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -2800000000.0)
     t_0
     (if (<= y.re 145000000000.0)
       (/ (fma x.re (/ y.re y.im) x.im) y.im)
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -2800000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 145000000000.0) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2800000000.0)
		tmp = t_0;
	elseif (y_46_re <= 145000000000.0)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2800000000.0], t$95$0, If[LessEqual[y$46$re, 145000000000.0], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.8e9 or 1.45e11 < y.re

   1. Initial program 48.7%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

   if -2.8e9 < y.re < 1.45e11

   1. Initial program 74.1%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.3

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

   4. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -3.7 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -3.7e-103) t_0 (if (<= y.re 2.7e-93) (/ x.im y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -3.7e-103) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-93) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.7e-103)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-93)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.7e-103], t$95$0, If[LessEqual[y$46$re, 2.7e-93], N[(x$46$im / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.6999999999999999e-103 or 2.7000000000000001e-93 < y.re

   1. Initial program 56.9%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 67.0

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

   4. Applied rewrites 67.0%

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

   if -3.6999999999999999e-103 < y.re < 2.7000000000000001e-93

   1. Initial program 71.2%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 71.5

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

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-39}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 0.018:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.5e+130)
   (/ x.re y.re)
   (if (<= y.re -8e-39)
     (* x.re (/ y.re (fma y.im y.im (* y.re y.re))))
     (if (<= y.re 0.018) (/ x.im y.im) (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.5e+130) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -8e-39) {
		tmp = x_46_re * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_re <= 0.018) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.5e+130)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -8e-39)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 0.018)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.5e+130], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -8e-39], N[(x$46$re * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.018], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.5e130 or 0.0179999999999999986 < y.re

   1. Initial program 44.6%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 67.5

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

   4. Applied rewrites 67.5%

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

   if -1.5e130 < y.re < -7.99999999999999943e-39

   1. Initial program 71.5%

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

   2. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

   if -7.99999999999999943e-39 < y.re < 0.0179999999999999986

   1. Initial program 73.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 9: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+157}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 0.018:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8e+157)
   (/ x.re y.re)
   (if (<= y.re -4e-103)
     (/ (fma y.im x.im (* y.re x.re)) (* y.re y.re))
     (if (<= y.re 0.018) (/ x.im y.im) (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8e+157) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -4e-103) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / (y_46_re * y_46_re);
	} else if (y_46_re <= 0.018) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8e+157)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -4e-103)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / Float64(y_46_re * y_46_re));
	elseif (y_46_re <= 0.018)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8e+157], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4e-103], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.018], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.99999999999999987e157 or 0.0179999999999999986 < y.re

   1. Initial program 43.6%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 67.8

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

   4. Applied rewrites 67.8%

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

   if -7.99999999999999987e157 < y.re < -3.99999999999999983e-103

   1. Initial program 72.7%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 55.6

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

   4. Applied rewrites 55.6%

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

   5. Taylor expanded in y.re around 0

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

      1. div-add N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 49.4

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

   7. Applied rewrites 49.4%

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

   if -3.99999999999999983e-103 < y.re < 0.0179999999999999986

   1. Initial program 72.7%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

Alternative 10: 63.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 0.018:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e-29)
   (/ x.re y.re)
   (if (<= y.re 0.018) (/ x.im y.im) (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-29) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 0.018) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.4d-29)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 0.018d0) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-29) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 0.018) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.4e-29:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 0.018:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e-29)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 0.018)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.4e-29)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 0.018)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.4e-29], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 0.018], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.39999999999999992e-29 or 0.0179999999999999986 < y.re

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

   if -2.39999999999999992e-29 < y.re < 0.0179999999999999986

   1. Initial program 73.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 64.6

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

   4. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 42.9% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.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_im / y_46_im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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_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_im / y_46_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im / y_46_im;
end

Wolfram

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

Derivation

1. Initial program 61.7%

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

2. Taylor expanded in y.re around 0

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

   1. lower-/.f64 42.9

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

4. Applied rewrites 42.9%

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

Reproduce

herbie shell --seed 2025095 
(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))))