_divideComplex, imaginary part

Percentage Accurate: 61.3% → 82.0%

Time: 4.2s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-164}:\\ \;\;\;\;-\frac{x.re - \frac{y.re \cdot x.im}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))
   (if (<= y.re -2.6e+129)
     t_1
     (if (<= y.re -6.2e-119)
       t_0
       (if (<= y.re 1.55e-164)
         (- (/ (- x.re (/ (* y.re x.im) y.im)) y.im))
         (if (<= y.re 1.25e+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_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.6e+129) {
		tmp = t_1;
	} else if (y_46_re <= -6.2e-119) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-164) {
		tmp = -((x_46_re - ((y_46_re * x_46_im) / y_46_im)) / y_46_im);
	} else if (y_46_re <= 1.25e+89) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im - ((y_46im / y_46re) * x_46re)) / y_46re
    if (y_46re <= (-2.6d+129)) then
        tmp = t_1
    else if (y_46re <= (-6.2d-119)) then
        tmp = t_0
    else if (y_46re <= 1.55d-164) then
        tmp = -((x_46re - ((y_46re * x_46im) / y_46im)) / y_46im)
    else if (y_46re <= 1.25d+89) then
        tmp = t_0
    else
        tmp = t_1
    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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.6e+129) {
		tmp = t_1;
	} else if (y_46_re <= -6.2e-119) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-164) {
		tmp = -((x_46_re - ((y_46_re * x_46_im) / y_46_im)) / y_46_im);
	} else if (y_46_re <= 1.25e+89) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -2.6e+129:
		tmp = t_1
	elif y_46_re <= -6.2e-119:
		tmp = t_0
	elif y_46_re <= 1.55e-164:
		tmp = -((x_46_re - ((y_46_re * x_46_im) / y_46_im)) / y_46_im)
	elif y_46_re <= 1.25e+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_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.6e+129)
		tmp = t_1;
	elseif (y_46_re <= -6.2e-119)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-164)
		tmp = Float64(-Float64(Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)) / y_46_im));
	elseif (y_46_re <= 1.25e+89)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.6e+129)
		tmp = t_1;
	elseif (y_46_re <= -6.2e-119)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-164)
		tmp = -((x_46_re - ((y_46_re * x_46_im) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 1.25e+89)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = 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] - N[(x$46$re * 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[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+129], t$95$1, If[LessEqual[y$46$re, -6.2e-119], t$95$0, If[LessEqual[y$46$re, 1.55e-164], (-N[(N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), If[LessEqual[y$46$re, 1.25e+89], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.60000000000000012e129 or 1.24999999999999996e89 < y.re

   1. Initial program 38.7%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.5

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 83.3

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

   6. Applied rewrites 83.3%

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

   if -2.60000000000000012e129 < y.re < -6.19999999999999956e-119 or 1.55e-164 < y.re < 1.24999999999999996e89

   1. Initial program 74.5%

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

   if -6.19999999999999956e-119 < y.re < 1.55e-164

   1. Initial program 69.5%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 24.5

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

   4. Applied rewrites 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 25.0

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

   6. Applied rewrites 25.0%

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

   7. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

         \[\leadsto -\frac{x.re - \frac{-1 \cdot \left(x.im \cdot y.re\right)}{-1 \cdot y.im}}{y.im} \]

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. frac-2neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 91.7

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

   9. Applied rewrites 91.7%

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

Alternative 2: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.28 \cdot 10^{-107}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+166}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= y.re -2.6e+129)
     (/ x.im y.re)
     (if (<= y.re -1.28e-107)
       (/ t_0 (* y.re y.re))
       (if (<= y.re 3e-105)
         (/ t_0 (* y.im y.im))
         (if (<= y.re 1.25e+166)
           (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))
           (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -2.6e+129) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.28e-107) {
		tmp = t_0 / (y_46_re * y_46_re);
	} else if (y_46_re <= 3e-105) {
		tmp = t_0 / (y_46_im * y_46_im);
	} else if (y_46_re <= 1.25e+166) {
		tmp = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.6e+129)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.28e-107)
		tmp = Float64(t_0 / Float64(y_46_re * y_46_re));
	elseif (y_46_re <= 3e-105)
		tmp = Float64(t_0 / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 1.25e+166)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(x_46_im / y_46_re);
	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 * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+129], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.28e-107], N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3e-105], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+166], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.60000000000000012e129 or 1.25e166 < y.re

   1. Initial program 33.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -2.60000000000000012e129 < y.re < -1.28e-107

   1. Initial program 74.0%

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

   2. Taylor expanded in y.re around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 49.4

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

   4. Applied rewrites 49.4%

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

   if -1.28e-107 < y.re < 3.0000000000000001e-105

   1. Initial program 71.1%

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

   2. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 66.3

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

   4. Applied rewrites 66.3%

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

   if 3.0000000000000001e-105 < y.re < 1.25e166

   1. Initial program 69.5%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 54.3

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

   4. Applied rewrites 54.3%

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

Alternative 3: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))))
   (if (<= y.re -9e+130)
     (/ x.im y.re)
     (if (<= y.re -4.8e-139)
       t_0
       (if (<= y.re 3e-105)
         (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
         (if (<= y.re 1.25e+166) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_re <= -9e+130) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.8e-139) {
		tmp = t_0;
	} else if (y_46_re <= 3e-105) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 1.25e+166) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_re <= -9e+130)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -4.8e-139)
		tmp = t_0;
	elseif (y_46_re <= 3e-105)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 1.25e+166)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im * 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, -9e+130], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.8e-139], t$95$0, If[LessEqual[y$46$re, 3e-105], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+166], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.00000000000000078e130 or 1.25e166 < y.re

   1. Initial program 32.6%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -9.00000000000000078e130 < y.re < -4.80000000000000029e-139 or 3.0000000000000001e-105 < y.re < 1.25e166

   1. Initial program 72.3%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 54.1

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

   4. Applied rewrites 54.1%

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

   if -4.80000000000000029e-139 < y.re < 3.0000000000000001e-105

   1. Initial program 70.2%

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

   2. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

Alternative 4: 64.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.56 \cdot 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-86}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))))
   (if (<= y.re -9e+130)
     (/ x.im y.re)
     (if (<= y.re -1.56e-220)
       t_0
       (if (<= y.re 1.45e-86)
         (/ (- x.re) y.im)
         (if (<= y.re 1.25e+166) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_re <= -9e+130) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.56e-220) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-86) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.25e+166) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_re <= -9e+130)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.56e-220)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-86)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.25e+166)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im * 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, -9e+130], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.56e-220], t$95$0, If[LessEqual[y$46$re, 1.45e-86], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+166], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.00000000000000078e130 or 1.25e166 < y.re

   1. Initial program 32.6%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -9.00000000000000078e130 < y.re < -1.5599999999999999e-220 or 1.45e-86 < y.re < 1.25e166

   1. Initial program 71.6%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

   if -1.5599999999999999e-220 < y.re < 1.45e-86

   1. Initial program 71.1%

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

   2. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x.re}{\color{blue}{y.im}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re}{\color{blue}{y.im}} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.2

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

   4. Applied rewrites 76.2%

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

Alternative 5: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{if}\;y.re \leq -70000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-84}:\\ \;\;\;\;-\frac{x.re - \frac{y.re \cdot x.im}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))
   (if (<= y.re -70000000000.0)
     t_0
     (if (<= y.re 1.3e-84)
       (- (/ (- x.re (/ (* y.re x.im) y.im)) y.im))
       (if (<= y.re 2.5e+89)
         (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -70000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.3e-84) {
		tmp = -((x_46_re - ((y_46_re * x_46_im) / y_46_im)) / y_46_im);
	} else if (y_46_re <= 2.5e+89) {
		tmp = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -70000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.3e-84)
		tmp = Float64(-Float64(Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)) / y_46_im));
	elseif (y_46_re <= 2.5e+89)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -70000000000.0], t$95$0, If[LessEqual[y$46$re, 1.3e-84], (-N[(N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), If[LessEqual[y$46$re, 2.5e+89], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7e10 or 2.49999999999999992e89 < y.re

   1. Initial program 45.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 78.3

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

   6. Applied rewrites 78.3%

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

   if -7e10 < y.re < 1.3e-84

   1. Initial program 72.9%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 30.3

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

   4. Applied rewrites 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 30.7

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

   6. Applied rewrites 30.7%

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

   7. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

         \[\leadsto -\frac{x.re - \frac{-1 \cdot \left(x.im \cdot y.re\right)}{-1 \cdot y.im}}{y.im} \]

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. frac-2neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

   if 1.3e-84 < y.re < 2.49999999999999992e89

   1. Initial program 74.0%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{if}\;y.re \leq -70000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))
   (if (<= y.re -70000000000.0)
     t_0
     (if (<= y.re 1.3e-84)
       (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
       (if (<= y.re 2.5e+89)
         (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -70000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.3e-84) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 2.5e+89) {
		tmp = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -70000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.3e-84)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 2.5e+89)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -70000000000.0], t$95$0, If[LessEqual[y$46$re, 1.3e-84], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+89], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7e10 or 2.49999999999999992e89 < y.re

   1. Initial program 45.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 78.3

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

   6. Applied rewrites 78.3%

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

   if -7e10 < y.re < 1.3e-84

   1. Initial program 72.9%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -1 \cdot x.re\right)}{y.im} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -1 \cdot x.re\right)}{y.im} \]

      6. mul-1-neg N/A

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

      7. lower-neg.f64 83.3

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

   4. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 81.2

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

   6. Applied rewrites 81.2%

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

   if 1.3e-84 < y.re < 2.49999999999999992e89

   1. Initial program 74.0%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

Alternative 7: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{if}\;y.re \leq -70000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))
   (if (<= y.re -70000000000.0)
     t_0
     (if (<= y.re 1.3e-84)
       (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
       (if (<= y.re 2.5e+89)
         (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -70000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.3e-84) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 2.5e+89) {
		tmp = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -70000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.3e-84)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 2.5e+89)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -70000000000.0], t$95$0, If[LessEqual[y$46$re, 1.3e-84], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+89], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7e10 or 2.49999999999999992e89 < y.re

   1. Initial program 45.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 78.3

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

   6. Applied rewrites 78.3%

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

   if -7e10 < y.re < 1.3e-84

   1. Initial program 72.9%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -1 \cdot x.re\right)}{y.im} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -1 \cdot x.re\right)}{y.im} \]

      6. mul-1-neg N/A

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

      7. lower-neg.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 1.3e-84 < y.re < 2.49999999999999992e89

   1. Initial program 74.0%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

Alternative 8: 67.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{if}\;y.re \leq -1.28 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))
   (if (<= y.re -1.28e-107)
     t_0
     (if (<= y.re 3e-105)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.re 2.5e+89)
         (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.28e-107) {
		tmp = t_0;
	} else if (y_46_re <= 3e-105) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 2.5e+89) {
		tmp = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.28e-107)
		tmp = t_0;
	elseif (y_46_re <= 3e-105)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 2.5e+89)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.28e-107], t$95$0, If[LessEqual[y$46$re, 3e-105], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+89], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.28e-107 or 2.49999999999999992e89 < y.re

   1. Initial program 51.4%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x.im - \frac{-1 \cdot \left(x.re \cdot y.im\right)}{-1 \cdot y.re}}{y.re} \]

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.2

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

   4. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 72.4

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

   6. Applied rewrites 72.4%

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

   if -1.28e-107 < y.re < 3.0000000000000001e-105

   1. Initial program 71.1%

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

   2. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 66.3

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

   4. Applied rewrites 66.3%

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

   if 3.0000000000000001e-105 < y.re < 2.49999999999999992e89

   1. Initial program 73.8%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

Alternative 9: 64.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -31500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -31500000000.0) t_0 (if (<= y.im 4.2e-36) (/ x.im y.re) t_0))))

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_im;
	double tmp;
	if (y_46_im <= -31500000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 4.2e-36) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-31500000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 4.2d-36) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    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 t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -31500000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 4.2e-36) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -31500000000.0:
		tmp = t_0
	elif y_46_im <= 4.2e-36:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -31500000000.0)
		tmp = t_0;
	elseif (y_46_im <= 4.2e-36)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -31500000000.0)
		tmp = t_0;
	elseif (y_46_im <= 4.2e-36)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -31500000000.0], t$95$0, If[LessEqual[y$46$im, 4.2e-36], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.15e10 or 4.19999999999999982e-36 < y.im

   1. Initial program 49.9%

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

   2. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x.re}{\color{blue}{y.im}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re}{\color{blue}{y.im}} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.9

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

   4. Applied rewrites 64.9%

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

   if -3.15e10 < y.im < 4.19999999999999982e-36

   1. Initial program 74.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 64.8

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

   4. Applied rewrites 64.8%

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

Alternative 10: 42.2% accurate, 3.2× speedup

Math

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

FPCore

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

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_re;
}

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_46re
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_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

2. Taylor expanded in y.re around inf

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

   1. lower-/.f64 42.2

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

4. Applied rewrites 42.2%

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

Reproduce

herbie shell --seed 2025095 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))