_divideComplex, imaginary part

Average Error: 26.3 → 17.1

Time: 8.6s

Precision: binary64

Cost: 2148

Math

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

↓

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.75 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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.re y.im)))
        (t_1
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.95e+123)
     t_0
     (if (<= y.im -1.32e-120)
       t_1
       (if (<= y.im 6.6e-192)
         (/ x.im y.re)
         (if (<= y.im 3.1e-73)
           t_1
           (if (<= y.im 1.05e-38)
             (/ x.im y.re)
             (if (<= y.im 6.8e+22)
               t_1
               (if (<= y.im 2.9e+50)
                 (/ x.im y.re)
                 (if (<= y.im 3.75e+65)
                   t_0
                   (if (<= y.im 1e+154) t_1 t_0)))))))))))

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_re / y_46_im);
	double t_1 = ((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 tmp;
	if (y_46_im <= -1.95e+123) {
		tmp = t_0;
	} else if (y_46_im <= -1.32e-120) {
		tmp = t_1;
	} else if (y_46_im <= 6.6e-192) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.1e-73) {
		tmp = t_1;
	} else if (y_46_im <= 1.05e-38) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 6.8e+22) {
		tmp = t_1;
	} else if (y_46_im <= 2.9e+50) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.75e+65) {
		tmp = t_0;
	} else if (y_46_im <= 1e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = -(x_46re / y_46im)
    t_1 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.95d+123)) then
        tmp = t_0
    else if (y_46im <= (-1.32d-120)) then
        tmp = t_1
    else if (y_46im <= 6.6d-192) then
        tmp = x_46im / y_46re
    else if (y_46im <= 3.1d-73) then
        tmp = t_1
    else if (y_46im <= 1.05d-38) then
        tmp = x_46im / y_46re
    else if (y_46im <= 6.8d+22) then
        tmp = t_1
    else if (y_46im <= 2.9d+50) then
        tmp = x_46im / y_46re
    else if (y_46im <= 3.75d+65) then
        tmp = t_0
    else if (y_46im <= 1d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	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_re / y_46_im);
	double t_1 = ((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 tmp;
	if (y_46_im <= -1.95e+123) {
		tmp = t_0;
	} else if (y_46_im <= -1.32e-120) {
		tmp = t_1;
	} else if (y_46_im <= 6.6e-192) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.1e-73) {
		tmp = t_1;
	} else if (y_46_im <= 1.05e-38) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 6.8e+22) {
		tmp = t_1;
	} else if (y_46_im <= 2.9e+50) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.75e+65) {
		tmp = t_0;
	} else if (y_46_im <= 1e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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_re / y_46_im)
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.95e+123:
		tmp = t_0
	elif y_46_im <= -1.32e-120:
		tmp = t_1
	elif y_46_im <= 6.6e-192:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 3.1e-73:
		tmp = t_1
	elif y_46_im <= 1.05e-38:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 6.8e+22:
		tmp = t_1
	elif y_46_im <= 2.9e+50:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 3.75e+65:
		tmp = t_0
	elif y_46_im <= 1e+154:
		tmp = t_1
	else:
		tmp = t_0
	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_re / y_46_im))
	t_1 = 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)))
	tmp = 0.0
	if (y_46_im <= -1.95e+123)
		tmp = t_0;
	elseif (y_46_im <= -1.32e-120)
		tmp = t_1;
	elseif (y_46_im <= 6.6e-192)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.1e-73)
		tmp = t_1;
	elseif (y_46_im <= 1.05e-38)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 6.8e+22)
		tmp = t_1;
	elseif (y_46_im <= 2.9e+50)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.75e+65)
		tmp = t_0;
	elseif (y_46_im <= 1e+154)
		tmp = t_1;
	else
		tmp = t_0;
	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_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -(x_46_re / y_46_im);
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.95e+123)
		tmp = t_0;
	elseif (y_46_im <= -1.32e-120)
		tmp = t_1;
	elseif (y_46_im <= 6.6e-192)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 3.1e-73)
		tmp = t_1;
	elseif (y_46_im <= 1.05e-38)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 6.8e+22)
		tmp = t_1;
	elseif (y_46_im <= 2.9e+50)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 3.75e+65)
		tmp = t_0;
	elseif (y_46_im <= 1e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = 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[(x$46$re / y$46$im), $MachinePrecision])}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$im, -1.95e+123], t$95$0, If[LessEqual[y$46$im, -1.32e-120], t$95$1, If[LessEqual[y$46$im, 6.6e-192], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e-73], t$95$1, If[LessEqual[y$46$im, 1.05e-38], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.8e+22], t$95$1, If[LessEqual[y$46$im, 2.9e+50], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.75e+65], t$95$0, If[LessEqual[y$46$im, 1e+154], t$95$1, t$95$0]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.94999999999999996e123 or 2.9e50 < y.im < 3.75000000000000003e65 or 1.00000000000000004e154 < y.im

   1. Initial program 42.3

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

   2. Taylor expanded in y.re around 0 15.4

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

   3. Simplified 15.4

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

      [Start] 15.4	\[ -1 \cdot \frac{x.re}{y.im} \]
      rational_best.json-simplify-2 [=>] 15.4	\[ \color{blue}{\frac{x.re}{y.im} \cdot -1} \]
      rational_best.json-simplify-12 [=>] 15.4	\[ \color{blue}{-\frac{x.re}{y.im}} \]

   if -1.94999999999999996e123 < y.im < -1.32000000000000004e-120 or 6.59999999999999978e-192 < y.im < 3.09999999999999969e-73 or 1.05000000000000006e-38 < y.im < 6.8e22 or 3.75000000000000003e65 < y.im < 1.00000000000000004e154

   1. Initial program 17.5

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

   if -1.32000000000000004e-120 < y.im < 6.59999999999999978e-192 or 3.09999999999999969e-73 < y.im < 1.05000000000000006e-38 or 6.8e22 < y.im < 2.9e50

   1. Initial program 22.6

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

   2. Taylor expanded in y.re around inf 18.4

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.75 \cdot 10^{+65}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 10^{+154}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.9
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.75 \cdot 10^{-33}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 2
Error	37.3
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))