_divideComplex, imaginary part

Percentage Accurate: 61.3% → 84.3%

Time: 6.7s

Alternatives: 9

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: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, \frac{y.re}{t\_0} \cdot x.im\right)\\ t_2 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma (- x.re) (/ y.im t_0) (* (/ y.re t_0) x.im)))
        (t_2 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -1.55e+120)
     t_2
     (if (<= y.re -3.2e-141)
       t_1
       (if (<= y.re 3.8e-71)
         (- (/ (/ (* y.re x.im) y.im) y.im) (/ x.re y.im))
         (if (<= y.re 2.7e+114) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(-x_46_re, (y_46_im / t_0), ((y_46_re / t_0) * x_46_im));
	double t_2 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.55e+120) {
		tmp = t_2;
	} else if (y_46_re <= -3.2e-141) {
		tmp = t_1;
	} else if (y_46_re <= 3.8e-71) {
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.7e+114) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(Float64(-x_46_re), Float64(y_46_im / t_0), Float64(Float64(y_46_re / t_0) * x_46_im))
	t_2 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.55e+120)
		tmp = t_2;
	elseif (y_46_re <= -3.2e-141)
		tmp = t_1;
	elseif (y_46_re <= 3.8e-71)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.7e+114)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x$46$re) * N[(y$46$im / t$95$0), $MachinePrecision] + N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.55e+120], t$95$2, If[LessEqual[y$46$re, -3.2e-141], t$95$1, If[LessEqual[y$46$re, 3.8e-71], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.7e+114], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.54999999999999987e120 or 2.7e114 < y.re

   1. Initial program 40.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 86.2%

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

   if -1.54999999999999987e120 < y.re < -3.2000000000000001e-141 or 3.79999999999999992e-71 < y.re < 2.7e114

   1. Initial program 77.0%

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

   4. Applied rewrites 83.7%

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

   if -3.2000000000000001e-141 < y.re < 3.79999999999999992e-71

   1. Initial program 63.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

Alternative 2: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -4.2e+39)
     t_0
     (if (<= y.re 1.45e-25)
       (- (/ (/ (* y.re x.im) y.im) y.im) (/ x.re y.im))
       (if (<= y.re 3.9e+95)
         (/ (fma (- x.re) y.im (* 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 * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -4.2e+39) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-25) {
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 3.9e+95) {
		tmp = fma(-x_46_re, y_46_im, (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(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -4.2e+39)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-25)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 3.9e+95)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(x_46_im * 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[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -4.2e+39], t$95$0, If[LessEqual[y$46$re, 1.45e-25], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.9e+95], N[(N[((-x$46$re) * y$46$im + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.1999999999999997e39 or 3.8999999999999997e95 < y.re

   1. Initial program 45.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 81.7%

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

   if -4.1999999999999997e39 < y.re < 1.45e-25

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 86.3%

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

   if 1.45e-25 < y.re < 3.8999999999999997e95

   1. Initial program 88.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 88.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 88.1%

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

Alternative 3: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -3.5e+43)
     t_0
     (if (<= y.re 1.45e-25)
       (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
       (if (<= y.re 3.9e+95)
         (/ (fma (- x.re) y.im (* 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 * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -3.5e+43) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-25) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.9e+95) {
		tmp = fma(-x_46_re, y_46_im, (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(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.5e+43)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-25)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.9e+95)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(x_46_im * 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[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+43], t$95$0, If[LessEqual[y$46$re, 1.45e-25], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.9e+95], N[(N[((-x$46$re) * y$46$im + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.5000000000000001e43 or 3.8999999999999997e95 < y.re

   1. Initial program 45.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 81.7%

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

   if -3.5000000000000001e43 < y.re < 1.45e-25

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 1.45e-25 < y.re < 3.8999999999999997e95

   1. Initial program 88.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 88.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 88.1%

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

Alternative 4: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -2.1e+35)
     t_0
     (if (<= y.re 4.6e-71)
       (/ (- x.re) y.im)
       (if (<= y.re 3.8e+78)
         (* (/ x.im (fma y.im y.im (* y.re 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 * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.1e+35) {
		tmp = t_0;
	} else if (y_46_re <= 4.6e-71) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 3.8e+78) {
		tmp = (x_46_im / fma(y_46_im, y_46_im, (y_46_re * 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(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.1e+35)
		tmp = t_0;
	elseif (y_46_re <= 4.6e-71)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 3.8e+78)
		tmp = Float64(Float64(x_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * 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[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e+35], t$95$0, If[LessEqual[y$46$re, 4.6e-71], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.8e+78], N[(N[(x$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.0999999999999999e35 or 3.7999999999999999e78 < y.re

   1. Initial program 47.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 80.0%

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

   if -2.0999999999999999e35 < y.re < 4.5999999999999997e-71

   1. Initial program 67.1%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if 4.5999999999999997e-71 < y.re < 3.7999999999999999e78

   1. Initial program 86.3%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.35e+104)
   (/ x.im y.re)
   (if (<= y.re 4.6e-71)
     (/ (- x.re) y.im)
     (if (<= y.re 1.7e+87)
       (* (/ x.im (fma y.im y.im (* y.re 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 tmp;
	if (y_46_re <= -1.35e+104) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 4.6e-71) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.7e+87) {
		tmp = (x_46_im / fma(y_46_im, y_46_im, (y_46_re * 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)
	tmp = 0.0
	if (y_46_re <= -1.35e+104)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 4.6e-71)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.7e+87)
		tmp = Float64(Float64(x_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * 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_] := If[LessEqual[y$46$re, -1.35e+104], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.6e-71], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+87], N[(N[(x$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.34999999999999992e104 or 1.7000000000000001e87 < y.re

   1. Initial program 43.3%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if -1.34999999999999992e104 < y.re < 4.5999999999999997e-71

   1. Initial program 67.4%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 4.5999999999999997e-71 < y.re < 1.7000000000000001e87

   1. Initial program 84.9%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.6

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

   5. Applied rewrites 68.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+43} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.5e+43) (not (<= y.re 8.5e-23)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (/ (- (/ (* x.im y.re) y.im) x.re) y.im)))

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 <= -3.5e+43) || !(y_46_re <= 8.5e-23)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	}
	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 <= (-3.5d+43)) .or. (.not. (y_46re <= 8.5d-23))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    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 <= -3.5e+43) || !(y_46_re <= 8.5e-23)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.5e+43) or not (y_46_re <= 8.5e-23):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.5e+43) || !(y_46_re <= 8.5e-23))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	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 <= -3.5e+43) || ~((y_46_re <= 8.5e-23)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.5e+43], N[Not[LessEqual[y$46$re, 8.5e-23]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.5000000000000001e43 or 8.4999999999999996e-23 < y.re

   1. Initial program 56.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 77.3%

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

   if -3.5000000000000001e43 < y.re < 8.4999999999999996e-23

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+43} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+43} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.5e+43) (not (<= y.re 8.5e-23)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (/ (- (* (/ y.re y.im) x.im) x.re) y.im)))

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 <= -3.5e+43) || !(y_46_re <= 8.5e-23)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im;
	}
	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 <= (-3.5d+43)) .or. (.not. (y_46re <= 8.5d-23))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = (((y_46re / y_46im) * x_46im) - x_46re) / y_46im
    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 <= -3.5e+43) || !(y_46_re <= 8.5e-23)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.5e+43) or not (y_46_re <= 8.5e-23):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.5e+43) || !(y_46_re <= 8.5e-23))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im);
	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 <= -3.5e+43) || ~((y_46_re <= 8.5e-23)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.5e+43], N[Not[LessEqual[y$46$re, 8.5e-23]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.5000000000000001e43 or 8.4999999999999996e-23 < y.re

   1. Initial program 56.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 77.3%

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

   if -3.5000000000000001e43 < y.re < 8.4999999999999996e-23

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 86.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+43} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+104} \lor \neg \left(y.re \leq 1.3 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.35e+104) (not (<= y.re 1.3e-24)))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

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 <= -1.35e+104) || !(y_46_re <= 1.3e-24)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	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 <= (-1.35d+104)) .or. (.not. (y_46re <= 1.3d-24))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    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 <= -1.35e+104) || !(y_46_re <= 1.3e-24)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.35e+104) or not (y_46_re <= 1.3e-24):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.35e+104) || !(y_46_re <= 1.3e-24))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	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 <= -1.35e+104) || ~((y_46_re <= 1.3e-24)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.35e+104], N[Not[LessEqual[y$46$re, 1.3e-24]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.34999999999999992e104 or 1.3e-24 < y.re

   1. Initial program 55.3%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -1.34999999999999992e104 < y.re < 1.3e-24

   1. Initial program 67.8%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 64.9

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

   5. Applied rewrites 64.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+104} \lor \neg \left(y.re \leq 1.3 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.0% 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 62.4%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 39.2

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

5. Applied rewrites 39.2%

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

Reproduce

herbie shell --seed 2024352 
(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))))