math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 96.3%

Time: 8.5s

Alternatives: 9

Speedup: 1.1×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

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

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

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

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

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

Alternative 1: 96.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.8 \cdot 10^{+150}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.8e+150)
   (* x.im (+ (- (* x.re x.re) (* x.im x.im)) (* (* x.re x.re) 2.0)))
   (* x.re (* x.re (* x.im 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.8e+150) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + ((x_46_re * x_46_re) * 2.0));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 7.8d+150) then
        tmp = x_46im * (((x_46re * x_46re) - (x_46im * x_46im)) + ((x_46re * x_46re) * 2.0d0))
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.8e+150) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + ((x_46_re * x_46_re) * 2.0));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 7.8e+150:
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + ((x_46_re * x_46_re) * 2.0))
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.8e+150)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) + Float64(Float64(x_46_re * x_46_re) * 2.0)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 7.8e+150)
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + ((x_46_re * x_46_re) * 2.0));
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7.8e+150], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.79999999999999981e150

   1. Initial program 80.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 80.7%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 80.7%

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

      2. *-commutative 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 80.7%

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

      2. distribute-lft-out 86.7%

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

      3. *-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 7.79999999999999981e150 < x.re

   1. Initial program 46.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 46.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 46.8%

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

      2. *-commutative 46.8%

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

      3. unpow2 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in x.re around 0 42.6%

      \[\leadsto \color{blue}{\left({x.re}^{2} \cdot x.im + -1 \cdot {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 42.6%

         \[\leadsto \left(\color{blue}{x.im \cdot {x.re}^{2}} + -1 \cdot {x.im}^{3}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      2. fma-def 42.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, {x.re}^{2}, -1 \cdot {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      3. mul-1-neg 42.6%

         \[\leadsto \mathsf{fma}\left(x.im, {x.re}^{2}, \color{blue}{-{x.im}^{3}}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      4. fma-neg 42.6%

         \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      5. unpow2 42.6%

         \[\leadsto \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} - {x.im}^{3}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

   7. Simplified 42.6%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right) - {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

   8. Taylor expanded in x.im around 0 67.6%

      \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]
   9. Step-by-step derivation

      1. unpow2 67.6%

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

      2. unpow2 67.6%

         \[\leadsto \left(2 \cdot \left(x.re \cdot x.re\right) + \color{blue}{x.re \cdot x.re}\right) \cdot x.im \]

      3. distribute-lft1-in 67.6%

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

      4. metadata-eval 67.6%

         \[\leadsto \left(\color{blue}{3} \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im \]

      5. *-commutative 67.6%

         \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot x.im \]

      6. associate-*r* 67.6%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      7. *-commutative 67.6%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(x.im \cdot 3\right)} \]

      8. associate-*l* 87.4%

         \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]

   10. Simplified 87.4%

       \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.8 \cdot 10^{+150}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 2: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) - x.im \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -1.8 \cdot 10^{+162}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (- (* x.re (* x.re 2.0)) (* x.im x.im)))))
   (if (<= x.im -1.8e+162)
     (* (+ x.re x.im) (* x.re x.im))
     (if (<= x.im -5.9e-64)
       t_0
       (if (<= x.im 2.6e-101)
         (* x.re (* x.re (* x.im 3.0)))
         (if (<= x.im 1.6e+146) t_0 (* x.im (* x.im (- x.im)))))))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re * (x_46_re * 2.0)) - (x_46_im * x_46_im));
	double tmp;
	if (x_46_im <= -1.8e+162) {
		tmp = (x_46_re + x_46_im) * (x_46_re * x_46_im);
	} else if (x_46_im <= -5.9e-64) {
		tmp = t_0;
	} else if (x_46_im <= 2.6e-101) {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	} else if (x_46_im <= 1.6e+146) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * ((x_46re * (x_46re * 2.0d0)) - (x_46im * x_46im))
    if (x_46im <= (-1.8d+162)) then
        tmp = (x_46re + x_46im) * (x_46re * x_46im)
    else if (x_46im <= (-5.9d-64)) then
        tmp = t_0
    else if (x_46im <= 2.6d-101) then
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    else if (x_46im <= 1.6d+146) then
        tmp = t_0
    else
        tmp = x_46im * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re * (x_46_re * 2.0)) - (x_46_im * x_46_im));
	double tmp;
	if (x_46_im <= -1.8e+162) {
		tmp = (x_46_re + x_46_im) * (x_46_re * x_46_im);
	} else if (x_46_im <= -5.9e-64) {
		tmp = t_0;
	} else if (x_46_im <= 2.6e-101) {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	} else if (x_46_im <= 1.6e+146) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	t_0 = x_46_im * ((x_46_re * (x_46_re * 2.0)) - (x_46_im * x_46_im))
	tmp = 0
	if x_46_im <= -1.8e+162:
		tmp = (x_46_re + x_46_im) * (x_46_re * x_46_im)
	elif x_46_im <= -5.9e-64:
		tmp = t_0
	elif x_46_im <= 2.6e-101:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	elif x_46_im <= 1.6e+146:
		tmp = t_0
	else:
		tmp = x_46_im * (x_46_im * -x_46_im)
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_re * Float64(x_46_re * 2.0)) - Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (x_46_im <= -1.8e+162)
		tmp = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * x_46_im));
	elseif (x_46_im <= -5.9e-64)
		tmp = t_0;
	elseif (x_46_im <= 2.6e-101)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	elseif (x_46_im <= 1.6e+146)
		tmp = t_0;
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * ((x_46_re * (x_46_re * 2.0)) - (x_46_im * x_46_im));
	tmp = 0.0;
	if (x_46_im <= -1.8e+162)
		tmp = (x_46_re + x_46_im) * (x_46_re * x_46_im);
	elseif (x_46_im <= -5.9e-64)
		tmp = t_0;
	elseif (x_46_im <= 2.6e-101)
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	elseif (x_46_im <= 1.6e+146)
		tmp = t_0;
	else
		tmp = x_46_im * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(N[(x$46$re * N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.8e+162], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -5.9e-64], t$95$0, If[LessEqual[x$46$im, 2.6e-101], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.6e+146], t$95$0, N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -1.79999999999999997e162

   1. Initial program 41.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   3. Applied egg-rr 17.2%

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

   4. Taylor expanded in x.re around inf 45.0%

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

   if -1.79999999999999997e162 < x.im < -5.89999999999999995e-64 or 2.6000000000000001e-101 < x.im < 1.6e146

   1. Initial program 98.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 98.5%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 98.5%

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

      2. *-commutative 98.5%

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

      3. unpow2 98.5%

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

   4. Simplified 98.5%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 98.5%

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

      2. distribute-lft-out 98.5%

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

      3. *-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in x.re around 0 90.1%

      \[\leadsto x.im \cdot \left(\color{blue}{-1 \cdot {x.im}^{2}} + \left(x.re \cdot x.re\right) \cdot 2\right) \]
   8. Step-by-step derivation

      1. unpow2 90.1%

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

      2. mul-1-neg 90.1%

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

      3. distribute-rgt-neg-in 90.1%

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

   9. Simplified 90.1%

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

   10. Taylor expanded in x.im around 0 90.1%

       \[\leadsto x.im \cdot \color{blue}{\left(2 \cdot {x.re}^{2} + -1 \cdot {x.im}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 90.1%

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

       2. *-commutative 90.1%

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

       3. associate-*r* 90.1%

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

       4. unpow2 90.1%

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

       5. mul-1-neg 90.1%

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

       6. sub-neg 90.1%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot 2\right) - x.im \cdot x.im\right)} \]

   12. Simplified 90.1%

       \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot 2\right) - x.im \cdot x.im\right)} \]

   if -5.89999999999999995e-64 < x.im < 2.6000000000000001e-101

   1. Initial program 88.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 88.2%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

      3. unpow2 88.3%

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

   4. Simplified 88.3%

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

   5. Taylor expanded in x.re around 0 88.3%

      \[\leadsto \color{blue}{\left({x.re}^{2} \cdot x.im + -1 \cdot {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \left(\color{blue}{x.im \cdot {x.re}^{2}} + -1 \cdot {x.im}^{3}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      2. fma-def 88.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, {x.re}^{2}, -1 \cdot {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      3. mul-1-neg 88.3%

         \[\leadsto \mathsf{fma}\left(x.im, {x.re}^{2}, \color{blue}{-{x.im}^{3}}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      4. fma-neg 88.3%

         \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      5. unpow2 88.3%

         \[\leadsto \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} - {x.im}^{3}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right) - {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

   8. Taylor expanded in x.im around 0 85.4%

      \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]
   9. Step-by-step derivation

      1. unpow2 85.4%

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

      2. unpow2 85.4%

         \[\leadsto \left(2 \cdot \left(x.re \cdot x.re\right) + \color{blue}{x.re \cdot x.re}\right) \cdot x.im \]

      3. distribute-lft1-in 85.4%

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

      4. metadata-eval 85.4%

         \[\leadsto \left(\color{blue}{3} \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im \]

      5. *-commutative 85.4%

         \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot x.im \]

      6. associate-*r* 85.4%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      7. *-commutative 85.4%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(x.im \cdot 3\right)} \]

      8. associate-*l* 96.9%

         \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]

   10. Simplified 96.9%

       \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]

   if 1.6e146 < x.im

   1. Initial program 39.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 39.5%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 39.5%

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

      2. *-commutative 39.5%

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

      3. unpow2 39.5%

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

   4. Simplified 39.5%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.5%

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

      2. distribute-lft-out 58.1%

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

      3. *-commutative 58.1%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in x.re around 0 76.7%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 76.7%

         \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. mul-1-neg 76.7%

         \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 76.7%

         \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   9. Simplified 76.7%

      \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.8 \cdot 10^{+162}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{+146}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 3: 69.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.45 \cdot 10^{+170} \lor \neg \left(x.re \leq 9.2 \cdot 10^{+179}\right) \land x.re \leq 4.2 \cdot 10^{+221}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re 1.45e+170) (and (not (<= x.re 9.2e+179)) (<= x.re 4.2e+221)))
   (* x.im (* x.im (- x.im)))
   (* x.re (* x.re x.im))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= 1.45e+170) || (!(x_46_re <= 9.2e+179) && (x_46_re <= 4.2e+221))) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= 1.45d+170) .or. (.not. (x_46re <= 9.2d+179)) .and. (x_46re <= 4.2d+221)) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46re * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= 1.45e+170) || (!(x_46_re <= 9.2e+179) && (x_46_re <= 4.2e+221))) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= 1.45e+170) or (not (x_46_re <= 9.2e+179) and (x_46_re <= 4.2e+221)):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * (x_46_re * x_46_im)
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= 1.45e+170) || (!(x_46_re <= 9.2e+179) && (x_46_re <= 4.2e+221)))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= 1.45e+170) || (~((x_46_re <= 9.2e+179)) && (x_46_re <= 4.2e+221)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, 1.45e+170], And[N[Not[LessEqual[x$46$re, 9.2e+179]], $MachinePrecision], LessEqual[x$46$re, 4.2e+221]]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.45e170 or 9.19999999999999976e179 < x.re < 4.20000000000000004e221

   1. Initial program 78.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 78.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 78.8%

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

      2. *-commutative 78.8%

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

      3. unpow2 78.8%

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

   4. Simplified 78.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 78.8%

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

      2. distribute-lft-out 84.6%

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

      3. *-commutative 84.6%

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

   6. Applied egg-rr 84.6%

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

   7. Taylor expanded in x.re around 0 64.3%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 64.3%

         \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. mul-1-neg 64.3%

         \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 64.3%

         \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   9. Simplified 64.3%

      \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if 1.45e170 < x.re < 9.19999999999999976e179 or 4.20000000000000004e221 < x.re

   1. Initial program 59.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around inf 89.0%

      \[\leadsto \color{blue}{{x.re}^{2}} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   3. Step-by-step derivation

      1. unpow2 89.0%

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

   4. Simplified 89.0%

      \[\leadsto \color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. *-commutative 89.0%

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

      2. *-un-lft-identity 89.0%

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

      3. *-un-lft-identity 89.0%

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

      4. distribute-rgt-out 89.0%

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

      5. metadata-eval 89.0%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.re \]

   6. Applied egg-rr 89.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.re \]
   7. Step-by-step derivation

      1. add-log-exp 89.0%

         \[\leadsto \color{blue}{\log \left(e^{\left(x.re \cdot x.re\right) \cdot x.im + \left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re}\right)} \]

      2. +-commutative 89.0%

         \[\leadsto \log \left(e^{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re + \left(x.re \cdot x.re\right) \cdot x.im}}\right) \]

      3. exp-sum 89.0%

         \[\leadsto \log \color{blue}{\left(e^{\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right)} \]

      4. *-commutative 89.0%

         \[\leadsto \log \left(e^{\color{blue}{x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      5. exp-prod 89.0%

         \[\leadsto \log \left(\color{blue}{{\left(e^{x.re}\right)}^{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      6. *-commutative 89.0%

         \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      7. count-2 89.0%

         \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      8. *-commutative 89.0%

         \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{x.im \cdot x.re}\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      9. add-sqr-sqrt 41.5%

         \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      10. sqrt-prod 77.3%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\sqrt{x.im \cdot x.im}} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      11. sqr-neg 77.3%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \sqrt{\color{blue}{\left(-x.im\right) \cdot \left(-x.im\right)}} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      12. sqrt-unprod 23.5%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\left(\sqrt{-x.im} \cdot \sqrt{-x.im}\right)} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      13. add-sqr-sqrt 53.7%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\left(-x.im\right)} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      14. cancel-sign-sub-inv 53.7%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{\left(x.re \cdot x.im - x.im \cdot x.re\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      15. *-commutative 53.7%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im - \color{blue}{x.re \cdot x.im}\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      16. +-inverses 89.0%

          \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{0}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      17. metadata-eval 89.0%

          \[\leadsto \log \left(\color{blue}{1} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

      18. *-un-lft-identity 89.0%

          \[\leadsto \log \color{blue}{\left(e^{\left(x.re \cdot x.re\right) \cdot x.im}\right)} \]

   8. Applied egg-rr 90.3%

      \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.45 \cdot 10^{+170} \lor \neg \left(x.re \leq 9.2 \cdot 10^{+179}\right) \land x.re \leq 4.2 \cdot 10^{+221}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -0.00038 \lor \neg \left(x.im \leq 0.00025\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -0.00038) (not (<= x.im 0.00025)))
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* x.im (* x.re x.re)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -0.00038) || !(x_46_im <= 0.00025)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-0.00038d0)) .or. (.not. (x_46im <= 0.00025d0))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -0.00038) || !(x_46_im <= 0.00025)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -0.00038) or not (x_46_im <= 0.00025):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -0.00038) || !(x_46_im <= 0.00025))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -0.00038) || ~((x_46_im <= 0.00025)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -0.00038], N[Not[LessEqual[x$46$im, 0.00025]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.8000000000000002e-4 or 2.5000000000000001e-4 < x.im

   1. Initial program 62.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 62.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 62.9%

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

      2. *-commutative 62.9%

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

      3. unpow2 62.9%

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

   4. Simplified 62.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 62.9%

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

      2. distribute-lft-out 75.0%

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

      3. *-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in x.re around 0 78.4%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 78.4%

         \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. mul-1-neg 78.4%

         \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 78.4%

         \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   9. Simplified 78.4%

      \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if -3.8000000000000002e-4 < x.im < 2.5000000000000001e-4

   1. Initial program 89.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 89.7%

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

      2. *-commutative 89.7%

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

      3. sub-neg 89.7%

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

      4. distribute-lft-in 89.7%

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

      5. associate-+r+ 89.7%

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

      6. distribute-rgt-neg-out 89.7%

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

      7. unsub-neg 89.7%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

          \[\leadsto x.re \cdot \left(\color{blue}{3} \cdot \left(x.im \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right) \]

      14. *-commutative 99.7%

          \[\leadsto x.re \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot 3\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

      15. *-commutative 99.7%

          \[\leadsto x.re \cdot \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right) \]

      16. associate-*r* 99.7%

          \[\leadsto x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

      17. cube-unmult 99.7%

          \[\leadsto x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - \color{blue}{{x.im}^{3}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}} \]

   4. Taylor expanded in x.re around inf 80.6%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   5. Step-by-step derivation

      1. *-commutative 80.6%

         \[\leadsto 3 \cdot \color{blue}{\left(x.im \cdot {x.re}^{2}\right)} \]

      2. unpow2 80.6%

         \[\leadsto 3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   6. Simplified 80.6%

      \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -0.00038 \lor \neg \left(x.im \leq 0.00025\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 5: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -0.00017 \lor \neg \left(x.im \leq 3.5 \cdot 10^{-5}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -0.00017) (not (<= x.im 3.5e-5)))
   (* x.im (* x.im (- x.im)))
   (* x.im (* x.re (* x.re 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -0.00017) || !(x_46_im <= 3.5e-5)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-0.00017d0)) .or. (.not. (x_46im <= 3.5d-5))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46im * (x_46re * (x_46re * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -0.00017) || !(x_46_im <= 3.5e-5)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -0.00017) or not (x_46_im <= 3.5e-5):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -0.00017) || !(x_46_im <= 3.5e-5))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_re * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -0.00017) || ~((x_46_im <= 3.5e-5)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -0.00017], N[Not[LessEqual[x$46$im, 3.5e-5]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.7e-4 or 3.4999999999999997e-5 < x.im

   1. Initial program 62.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 62.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 62.9%

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

      2. *-commutative 62.9%

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

      3. unpow2 62.9%

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

   4. Simplified 62.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 62.9%

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

      2. distribute-lft-out 75.0%

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

      3. *-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in x.re around 0 78.4%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 78.4%

         \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. mul-1-neg 78.4%

         \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 78.4%

         \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   9. Simplified 78.4%

      \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if -1.7e-4 < x.im < 3.4999999999999997e-5

   1. Initial program 89.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 89.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 89.6%

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

      2. *-commutative 89.6%

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

      3. unpow2 89.6%

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

   4. Simplified 89.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 89.6%

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

      2. distribute-lft-out 89.6%

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

      3. *-commutative 89.6%

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

   6. Applied egg-rr 89.6%

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

   7. Taylor expanded in x.im around 0 80.5%

      \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]
   8. Step-by-step derivation

      1. *-commutative 80.5%

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

      2. unpow2 80.5%

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

      3. +-commutative 80.5%

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

      4. unpow2 80.5%

         \[\leadsto x.im \cdot \left(x.re \cdot x.re + 2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      5. *-commutative 80.5%

         \[\leadsto x.im \cdot \left(x.re \cdot x.re + \color{blue}{\left(x.re \cdot x.re\right) \cdot 2}\right) \]

      6. associate-*r* 80.5%

         \[\leadsto x.im \cdot \left(x.re \cdot x.re + \color{blue}{x.re \cdot \left(x.re \cdot 2\right)}\right) \]

      7. *-rgt-identity 80.5%

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

      8. associate-*r* 80.5%

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

      9. distribute-lft-out 80.5%

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

      10. metadata-eval 80.5%

          \[\leadsto x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot \color{blue}{3}\right) \]

      11. associate-*l* 80.6%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot 3\right)\right)} \]

   9. Simplified 80.6%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -0.00017 \lor \neg \left(x.im \leq 3.5 \cdot 10^{-5}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 6: 80.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6 \cdot 10^{-7} \lor \neg \left(x.im \leq 0.00195\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6e-7) (not (<= x.im 0.00195)))
   (* x.im (* x.im (- x.im)))
   (* x.re (* x.re (* x.im 3.0)))))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6e-7) || !(x_46_im <= 0.00195)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-6d-7)) .or. (.not. (x_46im <= 0.00195d0))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6e-7) || !(x_46_im <= 0.00195)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6e-7) or not (x_46_im <= 0.00195):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -6e-7) || !(x_46_im <= 0.00195))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

x.re = abs(x.re)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -6e-7) || ~((x_46_im <= 0.00195)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -6e-7], N[Not[LessEqual[x$46$im, 0.00195]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5.9999999999999997e-7 or 0.0019499999999999999 < x.im

   1. Initial program 62.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 62.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 62.9%

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

      2. *-commutative 62.9%

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

      3. unpow2 62.9%

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

   4. Simplified 62.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 62.9%

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

      2. distribute-lft-out 75.0%

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

      3. *-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in x.re around 0 78.4%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 78.4%

         \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. mul-1-neg 78.4%

         \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 78.4%

         \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   9. Simplified 78.4%

      \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if -5.9999999999999997e-7 < x.im < 0.0019499999999999999

   1. Initial program 89.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 89.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 89.6%

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

      2. *-commutative 89.6%

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

      3. unpow2 89.6%

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

   4. Simplified 89.6%

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

   5. Taylor expanded in x.re around 0 89.7%

      \[\leadsto \color{blue}{\left({x.re}^{2} \cdot x.im + -1 \cdot {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 89.7%

         \[\leadsto \left(\color{blue}{x.im \cdot {x.re}^{2}} + -1 \cdot {x.im}^{3}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      2. fma-def 89.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, {x.re}^{2}, -1 \cdot {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      3. mul-1-neg 89.7%

         \[\leadsto \mathsf{fma}\left(x.im, {x.re}^{2}, \color{blue}{-{x.im}^{3}}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      4. fma-neg 89.7%

         \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

      5. unpow2 89.7%

         \[\leadsto \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} - {x.im}^{3}\right) + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

   7. Simplified 89.7%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right) - {x.im}^{3}\right)} + x.im \cdot \left(2 \cdot \left(x.re \cdot x.re\right)\right) \]

   8. Taylor expanded in x.im around 0 80.5%

      \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]
   9. Step-by-step derivation

      1. unpow2 80.5%

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

      2. unpow2 80.5%

         \[\leadsto \left(2 \cdot \left(x.re \cdot x.re\right) + \color{blue}{x.re \cdot x.re}\right) \cdot x.im \]

      3. distribute-lft1-in 80.5%

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

      4. metadata-eval 80.5%

         \[\leadsto \left(\color{blue}{3} \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im \]

      5. *-commutative 80.5%

         \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot x.im \]

      6. associate-*r* 80.6%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      7. *-commutative 80.6%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(x.im \cdot 3\right)} \]

      8. associate-*l* 90.6%

         \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]

   10. Simplified 90.6%

       \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6 \cdot 10^{-7} \lor \neg \left(x.im \leq 0.00195\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 7: 35.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ x.re \cdot \left(x.re \cdot x.im\right) \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* x.re (* x.re x.im)))

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_re * x_46_im);
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46re * (x_46re * x_46im)
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_re * x_46_im);
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return x_46_re * (x_46_re * x_46_im)

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(x_46_re * x_46_im))
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * (x_46_re * x_46_im);
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

2. Taylor expanded in x.re around inf 54.1%

   \[\leadsto \color{blue}{{x.re}^{2}} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
3. Step-by-step derivation

   1. unpow2 54.1%

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

4. Simplified 54.1%

   \[\leadsto \color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
5. Step-by-step derivation

   1. *-commutative 54.1%

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

   2. *-un-lft-identity 54.1%

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

   3. *-un-lft-identity 54.1%

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

   4. distribute-rgt-out 54.1%

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

   5. metadata-eval 54.1%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.re \]

6. Applied egg-rr 54.1%

   \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.re \]
7. Step-by-step derivation

   1. add-log-exp 36.8%

      \[\leadsto \color{blue}{\log \left(e^{\left(x.re \cdot x.re\right) \cdot x.im + \left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re}\right)} \]

   2. +-commutative 36.8%

      \[\leadsto \log \left(e^{\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re + \left(x.re \cdot x.re\right) \cdot x.im}}\right) \]

   3. exp-sum 36.8%

      \[\leadsto \log \color{blue}{\left(e^{\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right)} \]

   4. *-commutative 36.8%

      \[\leadsto \log \left(e^{\color{blue}{x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   5. exp-prod 36.4%

      \[\leadsto \log \left(\color{blue}{{\left(e^{x.re}\right)}^{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   6. *-commutative 36.4%

      \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   7. count-2 36.4%

      \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   8. *-commutative 36.4%

      \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{x.im \cdot x.re}\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   9. add-sqr-sqrt 27.1%

      \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   10. sqrt-prod 33.7%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\sqrt{x.im \cdot x.im}} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   11. sqr-neg 33.7%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \sqrt{\color{blue}{\left(-x.im\right) \cdot \left(-x.im\right)}} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   12. sqrt-unprod 24.2%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\left(\sqrt{-x.im} \cdot \sqrt{-x.im}\right)} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   13. add-sqr-sqrt 29.7%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im + \color{blue}{\left(-x.im\right)} \cdot x.re\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   14. cancel-sign-sub-inv 29.7%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{\left(x.re \cdot x.im - x.im \cdot x.re\right)}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   15. *-commutative 29.7%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\left(x.re \cdot x.im - \color{blue}{x.re \cdot x.im}\right)} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   16. +-inverses 36.7%

       \[\leadsto \log \left({\left(e^{x.re}\right)}^{\color{blue}{0}} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   17. metadata-eval 36.7%

       \[\leadsto \log \left(\color{blue}{1} \cdot e^{\left(x.re \cdot x.re\right) \cdot x.im}\right) \]

   18. *-un-lft-identity 36.7%

       \[\leadsto \log \color{blue}{\left(e^{\left(x.re \cdot x.re\right) \cdot x.im}\right)} \]

8. Applied egg-rr 39.6%

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

9. Final simplification 39.6%

   \[\leadsto x.re \cdot \left(x.re \cdot x.im\right) \]

Alternative 8: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ -3 \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 -3.0)

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -3.0d0
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return -3.0

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = -3.0;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 77.5%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative 77.5%

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

   2. *-commutative 77.5%

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

   3. sub-neg 77.5%

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

   4. distribute-lft-in 75.2%

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

   5. associate-+r+ 75.2%

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

   6. distribute-rgt-neg-out 75.2%

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

   7. unsub-neg 75.2%

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

   8. associate-*r* 80.7%

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

   9. distribute-rgt-out 80.7%

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

   10. *-commutative 80.7%

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

   11. count-2 80.7%

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

   12. distribute-lft1-in 80.7%

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

   13. metadata-eval 80.7%

       \[\leadsto x.re \cdot \left(\color{blue}{3} \cdot \left(x.im \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right) \]

   14. *-commutative 80.7%

       \[\leadsto x.re \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot 3\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

   15. *-commutative 80.7%

       \[\leadsto x.re \cdot \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right) \]

   16. associate-*r* 80.7%

       \[\leadsto x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

   17. cube-unmult 80.7%

       \[\leadsto x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - \color{blue}{{x.im}^{3}} \]

3. Simplified 80.7%

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}} \]

4. Taylor expanded in x.re around 0 80.7%

   \[\leadsto x.re \cdot \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - {x.im}^{3} \]
5. Step-by-step derivation

   1. associate-*r* 80.7%

      \[\leadsto x.re \cdot \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} - {x.im}^{3} \]

   2. *-commutative 80.7%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} - {x.im}^{3} \]

6. Simplified 80.7%

   \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} - {x.im}^{3} \]

7. Taylor expanded in x.re around 0 60.1%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]

10. Final simplification 2.7%

    \[\leadsto -3 \]

Alternative 9: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re = |x.re|\\ \\ 0.125 \end{array} \]

FPCore

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 0.125)

C

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	return 0.125;
}

Fortran

NOTE: x.re should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.125d0
end function

Java

x.re = Math.abs(x.re);
public static double code(double x_46_re, double x_46_im) {
	return 0.125;
}

Python

x.re = abs(x.re)
def code(x_46_re, x_46_im):
	return 0.125

Julia

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	return 0.125
end

MATLAB

x.re = abs(x.re)
function tmp = code(x_46_re, x_46_im)
	tmp = 0.125;
end

Wolfram

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := 0.125

Derivation

1. Initial program 77.5%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative 77.5%

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

   2. *-commutative 77.5%

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

   3. sub-neg 77.5%

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

   4. distribute-lft-in 75.2%

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

   5. associate-+r+ 75.2%

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

   6. distribute-rgt-neg-out 75.2%

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

   7. unsub-neg 75.2%

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

   8. associate-*r* 80.7%

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

   9. distribute-rgt-out 80.7%

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

   10. *-commutative 80.7%

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

   11. count-2 80.7%

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

   12. distribute-lft1-in 80.7%

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

   13. metadata-eval 80.7%

       \[\leadsto x.re \cdot \left(\color{blue}{3} \cdot \left(x.im \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right) \]

   14. *-commutative 80.7%

       \[\leadsto x.re \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot 3\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

   15. *-commutative 80.7%

       \[\leadsto x.re \cdot \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right) \]

   16. associate-*r* 80.7%

       \[\leadsto x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

   17. cube-unmult 80.7%

       \[\leadsto x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - \color{blue}{{x.im}^{3}} \]

3. Simplified 80.7%

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}} \]

4. Taylor expanded in x.re around 0 80.7%

   \[\leadsto x.re \cdot \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - {x.im}^{3} \]
5. Step-by-step derivation

   1. associate-*r* 80.7%

      \[\leadsto x.re \cdot \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} - {x.im}^{3} \]

   2. *-commutative 80.7%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} - {x.im}^{3} \]

6. Simplified 80.7%

   \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} - {x.im}^{3} \]
7. Step-by-step derivation

   1. sub-neg 80.7%

      \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right) + \left(-{x.im}^{3}\right)} \]

   2. flip3-+ 11.9%

      \[\leadsto \color{blue}{\frac{{\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right) + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right) \cdot \left(-{x.im}^{3}\right)\right)}} \]

   3. *-commutative 11.9%

      \[\leadsto \frac{{\left(x.re \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot 3\right)}\right)\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{\left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right) + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\right) \cdot \left(-{x.im}^{3}\right)\right)} \]

8. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{\frac{{\left(x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{\left(\left(x.im \cdot x.im\right) \cdot {x.re}^{4}\right) \cdot 9 + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\right) \cdot \left(-{x.im}^{3}\right)\right)}} \]

10. Simplified 2.8%

    \[\leadsto \color{blue}{0.125} \]

11. Final simplification 2.8%

    \[\leadsto 0.125 \]

Developer target: 91.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023238 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))