_divideComplex, imaginary part

Average Error: 26.5 → 11.1

Time: 2.3min

Precision: binary64

Cost: 1996

Math

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

↓

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := \frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ t_2 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4.7 \cdot 10^{-152}:\\ \;\;\;\;\frac{\begin{array}{l} \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{1}{\frac{1}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}}{t_2}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+92}:\\ \;\;\;\;\frac{t_0}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im)))
        (t_1 (/ (- (* (/ x.im y.im) y.re) x.re) y.im))
        (t_2 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.im -1.4e+121)
     t_1
     (if (<= y.im -4.7e-152)
       (/ (if (!= t_0 0.0) (/ 1.0 (/ 1.0 t_0)) t_0) t_2)
       (if (<= y.im 1.4e-164)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 7e+92) (/ t_0 t_2) t_1))))))

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));
}

↓

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 t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	double t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -1.4e+121) {
		tmp = t_1;
	} else if (y_46_im <= -4.7e-152) {
		double tmp_1;
		if (t_0 != 0.0) {
			tmp_1 = 1.0 / (1.0 / t_0);
		} else {
			tmp_1 = t_0;
		}
		tmp = tmp_1 / t_2;
	} else if (y_46_im <= 1.4e-164) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 7e+92) {
		tmp = t_0 / t_2;
	} else {
		tmp = t_1;
	}
	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_46im * y_46re) - (x_46re * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    t_0 = (x_46im * y_46re) - (x_46re * y_46im)
    t_1 = (((x_46im / y_46im) * y_46re) - x_46re) / y_46im
    t_2 = (y_46re * y_46re) + (y_46im * y_46im)
    if (y_46im <= (-1.4d+121)) then
        tmp = t_1
    else if (y_46im <= (-4.7d-152)) then
        if (t_0 /= 0.0d0) then
            tmp_1 = 1.0d0 / (1.0d0 / t_0)
        else
            tmp_1 = t_0
        end if
        tmp = tmp_1 / t_2
    else if (y_46im <= 1.4d-164) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 7d+92) then
        tmp = t_0 / t_2
    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) {
	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));
}

↓

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);
	double t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	double t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -1.4e+121) {
		tmp = t_1;
	} else if (y_46_im <= -4.7e-152) {
		double tmp_1;
		if (t_0 != 0.0) {
			tmp_1 = 1.0 / (1.0 / t_0);
		} else {
			tmp_1 = t_0;
		}
		tmp = tmp_1 / t_2;
	} else if (y_46_im <= 1.4e-164) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 7e+92) {
		tmp = t_0 / t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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))

↓

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)
	t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im
	t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -1.4e+121:
		tmp = t_1
	elif y_46_im <= -4.7e-152:
		tmp_1 = 0
		if t_0 != 0.0:
			tmp_1 = 1.0 / (1.0 / t_0)
		else:
			tmp_1 = t_0
		tmp = tmp_1 / t_2
	elif y_46_im <= 1.4e-164:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 7e+92:
		tmp = t_0 / t_2
	else:
		tmp = t_1
	return tmp

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

↓

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))
	t_1 = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im)
	t_2 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.4e+121)
		tmp = t_1;
	elseif (y_46_im <= -4.7e-152)
		tmp_1 = 0.0
		if (t_0 != 0.0)
			tmp_1 = Float64(1.0 / Float64(1.0 / t_0));
		else
			tmp_1 = t_0;
		end
		tmp = Float64(tmp_1 / t_2);
	elseif (y_46_im <= 1.4e-164)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 7e+92)
		tmp = Float64(t_0 / t_2);
	else
		tmp = t_1;
	end
	return tmp
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

↓

function tmp_3 = 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);
	t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.4e+121)
		tmp = t_1;
	elseif (y_46_im <= -4.7e-152)
		tmp_2 = 0.0;
		if (t_0 ~= 0.0)
			tmp_2 = 1.0 / (1.0 / t_0);
		else
			tmp_2 = t_0;
		end
		tmp = tmp_2 / t_2;
	elseif (y_46_im <= 1.4e-164)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 7e+92)
		tmp = t_0 / t_2;
	else
		tmp = t_1;
	end
	tmp_3 = tmp;
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]

↓

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]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.4e+121], t$95$1, If[LessEqual[y$46$im, -4.7e-152], N[(If[Unequal[t$95$0, 0.0], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$0] / t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 1.4e-164], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7e+92], N[(t$95$0 / t$95$2), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.40000000000000003e121 or 6.99999999999999972e92 < y.im

   1. Initial program 40.7

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

   2. Taylor expanded in y.re around 0 41.5

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

   3. Simplified 41.5

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

   4. Applied egg-rr 28.5

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

   5. Taylor expanded in x.re around 0 9.7

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

   6. Applied egg-rr 9.1

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

   if -1.40000000000000003e121 < y.im < -4.70000000000000012e-152

   1. Initial program 16.3

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

   2. Applied egg-rr 16.5

      \[\leadsto \frac{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x.im \cdot y.re - x.re \cdot y.im \ne 0:\\ \;\;\;\;\frac{1}{\frac{1}{x.im \cdot y.re - x.re \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re - x.re \cdot y.im\\ } \end{array}}}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -4.70000000000000012e-152 < y.im < 1.4000000000000001e-164

   1. Initial program 24.5

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

   2. Taylor expanded in y.re around inf 10.2

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

   3. Simplified 13.7

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

   4. Taylor expanded in x.im around 0 9.6

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

   5. Simplified 5.4

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

   if 1.4000000000000001e-164 < y.im < 6.99999999999999972e92

   1. Initial program 14.8

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

Alternatives

Alternative 1
Error	11.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.4
Cost	1104

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	20.5
Cost	1104

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.0
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.6
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{y.im} \cdot x.im - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.55 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.0
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{y.im} \cdot x.im - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{y.re} \cdot y.im\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]

Alternative 7
Error	23.9
Cost	784

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.32 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 8
Error	37.9
Cost	192

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

Reproduce

herbie shell --seed 2023033 
(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))))