_divideComplex, real part

Average Error: 26.5 → 15.8

Time: 23.6s

Precision: binary64

Cost: 7172

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.im \leq -3.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{1}{y.re \cdot x.re + y.im \cdot x.im} \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

FPCore

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

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.im -8.5e+102)
     (+ (/ x.im y.im) (* y.re (/ x.re (pow y.im 2.0))))
     (if (<= y.im -3.3e-127)
       (* (/ 1.0 t_0) (+ (* x.re y.re) (* x.im y.im)))
       (if (<= y.im 1.65e-105)
         (/ x.re y.re)
         (if (<= y.im 4e+136)
           (/ 1.0 (* (/ 1.0 (+ (* y.re x.re) (* y.im x.im))) t_0))
           (/ x.im y.im)))))))

C

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

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -8.5e+102) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_im <= -3.3e-127) {
		tmp = (1.0 / t_0) * ((x_46_re * y_46_re) + (x_46_im * y_46_im));
	} else if (y_46_im <= 1.65e-105) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4e+136) {
		tmp = 1.0 / ((1.0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im))) * t_0);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    if (y_46im <= (-8.5d+102)) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im ** 2.0d0)))
    else if (y_46im <= (-3.3d-127)) then
        tmp = (1.0d0 / t_0) * ((x_46re * y_46re) + (x_46im * y_46im))
    else if (y_46im <= 1.65d-105) then
        tmp = x_46re / y_46re
    else if (y_46im <= 4d+136) then
        tmp = 1.0d0 / ((1.0d0 / ((y_46re * x_46re) + (y_46im * x_46im))) * t_0)
    else
        tmp = x_46im / 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) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -8.5e+102) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / Math.pow(y_46_im, 2.0)));
	} else if (y_46_im <= -3.3e-127) {
		tmp = (1.0 / t_0) * ((x_46_re * y_46_re) + (x_46_im * y_46_im));
	} else if (y_46_im <= 1.65e-105) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4e+136) {
		tmp = 1.0 / ((1.0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im))) * t_0);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

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

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -8.5e+102:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / math.pow(y_46_im, 2.0)))
	elif y_46_im <= -3.3e-127:
		tmp = (1.0 / t_0) * ((x_46_re * y_46_re) + (x_46_im * y_46_im))
	elif y_46_im <= 1.65e-105:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 4e+136:
		tmp = 1.0 / ((1.0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im))) * t_0)
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

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

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -8.5e+102)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_im <= -3.3e-127)
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)));
	elseif (y_46_im <= 1.65e-105)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 4e+136)
		tmp = Float64(1.0 / Float64(Float64(1.0 / Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im))) * t_0));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -8.5e+102)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im ^ 2.0)));
	elseif (y_46_im <= -3.3e-127)
		tmp = (1.0 / t_0) * ((x_46_re * y_46_re) + (x_46_im * y_46_im));
	elseif (y_46_im <= 1.65e-105)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 4e+136)
		tmp = 1.0 / ((1.0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im))) * t_0);
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.5e+102], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.3e-127], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.65e-105], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4e+136], N[(1.0 / N[(N[(1.0 / N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.4999999999999996e102

   1. Initial program 41.1

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

   2. Taylor expanded in y.re around 0 15.4

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

   3. Simplified 13.9

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

      [Start] 15.4	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      rational.json-simplify-1 [=>] 15.4	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      rational.json-simplify-49 [=>] 13.9	\[ \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{x.re}{{y.im}^{2}}} \]

   if -8.4999999999999996e102 < y.im < -3.29999999999999981e-127

   1. Initial program 17.4

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

   2. Applied egg-rr 17.5

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

   if -3.29999999999999981e-127 < y.im < 1.6499999999999999e-105

   1. Initial program 22.0

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

   2. Taylor expanded in y.re around inf 15.8

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

   if 1.6499999999999999e-105 < y.im < 4.00000000000000023e136

   1. Initial program 17.0

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

   2. Applied egg-rr 17.1

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

   3. Applied egg-rr 17.2

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

   4. Applied egg-rr 17.4

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

   if 4.00000000000000023e136 < y.im

   1. Initial program 42.3

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

   2. Taylor expanded in y.re around 0 13.9

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.im \leq -3.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{y.re \cdot y.re + y.im \cdot y.im} \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{1}{y.re \cdot x.re + y.im \cdot x.im} \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.9
Cost	7172

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.im \leq -2.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{1}{y.re \cdot x.re + y.im \cdot x.im} \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 2
Error	16.2
Cost	1744

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{1}{y.re \cdot x.re + y.im \cdot x.im} \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 3
Error	16.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 4
Error	16.2
Cost	1488

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 5
Error	21.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.45 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 6
Error	21.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 7
Error	22.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \cdot y.im\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	37.4
Cost	192

\[\frac{x.im}{y.im} \]

Reproduce

herbie shell --seed 2023065 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))