math.cube on complex, real part

Percentage Accurate: 82.4% → 99.2%

Time: 7.8s

Alternatives: 11

Speedup: 1.1×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -2e+121)
   (* (* x.re (+ x.re -27.0)) (+ x.re x.im))
   (if (<= x.re 5e+98)
     (fma (* x.re x.im) (* x.im -3.0) (pow x.re 3.0))
     (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2e+121) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else if (x_46_re <= 5e+98) {
		tmp = fma((x_46_re * x_46_im), (x_46_im * -3.0), pow(x_46_re, 3.0));
	} else {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -2e+121)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re + -27.0)) * Float64(x_46_re + x_46_im));
	elseif (x_46_re <= 5e+98)
		tmp = fma(Float64(x_46_re * x_46_im), Float64(x_46_im * -3.0), (x_46_re ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -2e+121], N[(N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e+98], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.00000000000000007e121

   1. Initial program 57.6%

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

      1. difference-of-squares 66.7%

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

   3. Applied egg-rr 66.7%

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

   5. Simplified 57.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 42.4%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 42.4%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 42.4%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 51.5%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 51.5%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 51.5%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 51.5%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 78.8%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 78.8%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 90.9%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.im around 0 42.4%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]

   if -2.00000000000000007e121 < x.re < 4.9999999999999998e98

   1. Initial program 92.7%

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

   3. Simplified 92.8%

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

      1. associate-*r* 92.9%

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

      2. associate-*l* 92.8%

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

      3. +-commutative 92.8%

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

      4. associate-*r* 99.7%

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

      5. associate-*r* 99.8%

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

      6. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 4.9999999999999998e98 < x.re

   1. Initial program 58.3%

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

      1. difference-of-squares 66.7%

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

      2. *-commutative 66.7%

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

   3. Applied egg-rr 66.7%

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

      1. expm1-log1p-u 36.1%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.im \]

      2. expm1-udef 36.1%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.im \]

      3. *-commutative 36.1%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      4. flip-+ 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \cdot x.im \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      6. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.im \]

   5. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.im \]

   7. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+98}:\\ \;\;\;\;{x.re}^{3} + -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -2e+121)
   (* (* x.re (+ x.re -27.0)) (+ x.re x.im))
   (if (<= x.re 5e+98)
     (+ (pow x.re 3.0) (* -3.0 (* x.im (* x.re x.im))))
     (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2e+121) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else if (x_46_re <= 5e+98) {
		tmp = pow(x_46_re, 3.0) + (-3.0 * (x_46_im * (x_46_re * x_46_im)));
	} else {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= (-2d+121)) then
        tmp = (x_46re * (x_46re + (-27.0d0))) * (x_46re + x_46im)
    else if (x_46re <= 5d+98) then
        tmp = (x_46re ** 3.0d0) + ((-3.0d0) * (x_46im * (x_46re * x_46im)))
    else
        tmp = (x_46re * ((x_46re + x_46im) * (x_46re - x_46im))) - (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2e+121) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else if (x_46_re <= 5e+98) {
		tmp = Math.pow(x_46_re, 3.0) + (-3.0 * (x_46_im * (x_46_re * x_46_im)));
	} else {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -2e+121:
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im)
	elif x_46_re <= 5e+98:
		tmp = math.pow(x_46_re, 3.0) + (-3.0 * (x_46_im * (x_46_re * x_46_im)))
	else:
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -2e+121)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re + -27.0)) * Float64(x_46_re + x_46_im));
	elseif (x_46_re <= 5e+98)
		tmp = Float64((x_46_re ^ 3.0) + Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -2e+121)
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	elseif (x_46_re <= 5e+98)
		tmp = (x_46_re ^ 3.0) + (-3.0 * (x_46_im * (x_46_re * x_46_im)));
	else
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -2e+121], N[(N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e+98], N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.00000000000000007e121

   1. Initial program 57.6%

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

      1. difference-of-squares 66.7%

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

   3. Applied egg-rr 66.7%

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

   5. Simplified 57.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 42.4%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 42.4%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 42.4%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 51.5%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 51.5%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 51.5%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 51.5%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 78.8%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 78.8%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 90.9%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.im around 0 42.4%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]

   if -2.00000000000000007e121 < x.re < 4.9999999999999998e98

   1. Initial program 92.7%

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

   3. Simplified 92.8%

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

      1. add-sqr-sqrt 66.9%

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

      2. pow2 66.9%

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

      3. *-commutative 66.9%

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

      4. sqrt-prod 47.7%

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

      5. sqrt-prod 25.5%

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

      6. add-sqr-sqrt 51.1%

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

   5. Applied egg-rr 51.1%

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

      1. unpow2 51.1%

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

      2. *-commutative 51.1%

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

      3. associate-*r* 51.2%

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

      4. associate-*r* 51.2%

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

      5. add-sqr-sqrt 99.7%

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

      6. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 4.9999999999999998e98 < x.re

   1. Initial program 58.3%

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

      1. difference-of-squares 66.7%

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

      2. *-commutative 66.7%

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

   3. Applied egg-rr 66.7%

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

      1. expm1-log1p-u 36.1%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.im \]

      2. expm1-udef 36.1%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.im \]

      3. *-commutative 36.1%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      4. flip-+ 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \cdot x.im \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      6. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.im \]

   5. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.im \]

   7. Simplified 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+98}:\\ \;\;\;\;{x.re}^{3} + -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 3: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_0 - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* (+ x.re x.im) (- x.re x.im)))))
   (if (<=
        (-
         (* x.re (- (* x.re x.re) (* x.im x.im)))
         (* x.im (+ (* x.re x.im) (* x.re x.im))))
        5e+306)
     (- t_0 (* x.im (* (* x.re x.im) 2.0)))
     (- t_0 (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 5e+306) {
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	} else {
		tmp = t_0 - (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im 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_46re * ((x_46re + x_46im) * (x_46re - x_46im))
    if (((x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))) <= 5d+306) then
        tmp = t_0 - (x_46im * ((x_46re * x_46im) * 2.0d0))
    else
        tmp = t_0 - (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 5e+306) {
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	} else {
		tmp = t_0 - (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 5e+306:
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0))
	else:
		tmp = t_0 - (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= 5e+306)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * 2.0)));
	else
		tmp = Float64(t_0 - Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	tmp = 0.0;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 5e+306)
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	else
		tmp = t_0 - (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+306], N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.99999999999999993e306

   1. Initial program 96.2%

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

      1. difference-of-squares 96.2%

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

      2. *-commutative 96.2%

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

   3. Applied egg-rr 96.2%

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

      1. *-un-lft-identity 96.2%

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

      2. *-commutative 96.2%

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

      3. *-un-lft-identity 96.2%

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

      4. distribute-rgt-out 96.2%

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

      5. metadata-eval 96.2%

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

   5. Applied egg-rr 96.2%

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

   if 4.99999999999999993e306 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 50.6%

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

      1. difference-of-squares 59.0%

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

      2. *-commutative 59.0%

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

   3. Applied egg-rr 59.0%

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

      1. expm1-log1p-u 29.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.im \]

      2. expm1-udef 29.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.im \]

      3. *-commutative 29.7%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      4. flip-+ 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \cdot x.im \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      6. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.im \]

   5. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.im \]

   7. Simplified 90.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 4: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.5 \cdot 10^{-86} \lor \neg \left(x.re \leq 1.2 \cdot 10^{-18}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(\left(-x.re\right) - x.re \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.5e-86) (not (<= x.re 1.2e-18)))
   (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0))
   (* (* x.im x.im) (- (- x.re) (* x.re 2.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.5e-86) || !(x_46_re <= 1.2e-18)) {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	} else {
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= (-2.5d-86)) .or. (.not. (x_46re <= 1.2d-18))) then
        tmp = (x_46re * ((x_46re + x_46im) * (x_46re - x_46im))) - (x_46im * (-3.0d0))
    else
        tmp = (x_46im * x_46im) * (-x_46re - (x_46re * 2.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.5e-86) || !(x_46_re <= 1.2e-18)) {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	} else {
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2.5e-86) or not (x_46_re <= 1.2e-18):
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0)
	else:
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -2.5e-86) || !(x_46_re <= 1.2e-18))
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(Float64(-x_46_re) - Float64(x_46_re * 2.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -2.5e-86) || ~((x_46_re <= 1.2e-18)))
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	else
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2.5e-86], N[Not[LessEqual[x$46$re, 1.2e-18]], $MachinePrecision]], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[((-x$46$re) - N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.4999999999999999e-86 or 1.19999999999999997e-18 < x.re

   1. Initial program 77.4%

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

      1. difference-of-squares 81.6%

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

      2. *-commutative 81.6%

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

   3. Applied egg-rr 81.6%

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

      1. expm1-log1p-u 52.2%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.im \]

      2. expm1-udef 50.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.im \]

      3. *-commutative 50.8%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      4. flip-+ 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \cdot x.im \]

      5. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \cdot x.im \]

      6. +-inverses 0.0%

         \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.im \]

   5. Applied egg-rr 0.0%

      \[\leadsto \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.im \]

   7. Simplified 86.5%

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

   if -2.4999999999999999e-86 < x.re < 1.19999999999999997e-18

   1. Initial program 90.8%

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

   2. Taylor expanded in x.im around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.5 \cdot 10^{-86} \lor \neg \left(x.re \leq 1.2 \cdot 10^{-18}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(\left(-x.re\right) - x.re \cdot 2\right)\\ \end{array} \]

Alternative 5: 77.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5.2 \cdot 10^{+44} \lor \neg \left(x.re \leq 5.5 \cdot 10^{+55}\right):\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(\left(-x.re\right) - x.re \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -5.2e+44) (not (<= x.re 5.5e+55)))
   (* (* x.re (+ x.re -27.0)) (+ x.re x.im))
   (* (* x.im x.im) (- (- x.re) (* x.re 2.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5.2e+44) || !(x_46_re <= 5.5e+55)) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else {
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= (-5.2d+44)) .or. (.not. (x_46re <= 5.5d+55))) then
        tmp = (x_46re * (x_46re + (-27.0d0))) * (x_46re + x_46im)
    else
        tmp = (x_46im * x_46im) * (-x_46re - (x_46re * 2.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5.2e+44) || !(x_46_re <= 5.5e+55)) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else {
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -5.2e+44) or not (x_46_re <= 5.5e+55):
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im)
	else:
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -5.2e+44) || !(x_46_re <= 5.5e+55))
		tmp = Float64(Float64(x_46_re * Float64(x_46_re + -27.0)) * Float64(x_46_re + x_46_im));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(Float64(-x_46_re) - Float64(x_46_re * 2.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -5.2e+44) || ~((x_46_re <= 5.5e+55)))
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	else
		tmp = (x_46_im * x_46_im) * (-x_46_re - (x_46_re * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -5.2e+44], N[Not[LessEqual[x$46$re, 5.5e+55]], $MachinePrecision]], N[(N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[((-x$46$re) - N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.1999999999999998e44 or 5.5000000000000004e55 < x.re

   1. Initial program 68.1%

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

      1. difference-of-squares 74.7%

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

   3. Applied egg-rr 74.7%

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

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 53.8%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 53.8%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 53.8%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 60.4%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 60.4%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 60.4%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 60.4%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 79.1%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 79.1%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 90.1%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 90.1%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.im around 0 53.8%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]

   11. Simplified 90.1%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]

   if -5.1999999999999998e44 < x.re < 5.5000000000000004e55

   1. Initial program 91.8%

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

   2. Taylor expanded in x.im around inf 71.7%

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

      1. unpow2 71.7%

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

   4. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.2 \cdot 10^{+44} \lor \neg \left(x.re \leq 5.5 \cdot 10^{+55}\right):\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(\left(-x.re\right) - x.re \cdot 2\right)\\ \end{array} \]

Alternative 6: 54.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.9 \cdot 10^{+222}:\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.9e+222)
   (* (* x.re (+ x.re -27.0)) (+ x.re x.im))
   (* -27.0 (* x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.9e+222) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= 2.9d+222) then
        tmp = (x_46re * (x_46re + (-27.0d0))) * (x_46re + x_46im)
    else
        tmp = (-27.0d0) * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.9e+222) {
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.9e+222:
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im)
	else:
		tmp = -27.0 * (x_46_re * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.9e+222)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re + -27.0)) * Float64(x_46_re + x_46_im));
	else
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.9e+222)
		tmp = (x_46_re * (x_46_re + -27.0)) * (x_46_re + x_46_im);
	else
		tmp = -27.0 * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.9e+222], N[(N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.89999999999999981e222

   1. Initial program 83.9%

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

      1. difference-of-squares 85.6%

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

   3. Applied egg-rr 85.6%

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

   5. Simplified 51.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 36.1%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 36.1%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 36.1%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 38.6%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 38.6%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 38.6%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 38.6%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 45.7%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 45.7%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 49.5%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.im around 0 36.1%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]

   11. Simplified 49.5%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]

   if 2.89999999999999981e222 < x.im

   1. Initial program 75.4%

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

      1. difference-of-squares 87.9%

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

   3. Applied egg-rr 87.9%

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 20.7%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 20.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 20.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 20.7%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 20.7%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 20.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 20.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 20.7%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 20.7%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 27.0%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 27.0%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.re around 0 58.2%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

   11. Simplified 58.2%

       \[\leadsto \color{blue}{-27 \cdot \left(x.re \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.9 \cdot 10^{+222}:\\ \;\;\;\;\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 7: 54.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.6 \cdot 10^{+221}:\\ \;\;\;\;\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 3.6e+221)
   (* (+ x.re -27.0) (* x.re (+ x.re x.im)))
   (* -27.0 (* x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.6e+221) {
		tmp = (x_46_re + -27.0) * (x_46_re * (x_46_re + x_46_im));
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= 3.6d+221) then
        tmp = (x_46re + (-27.0d0)) * (x_46re * (x_46re + x_46im))
    else
        tmp = (-27.0d0) * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.6e+221) {
		tmp = (x_46_re + -27.0) * (x_46_re * (x_46_re + x_46_im));
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 3.6e+221:
		tmp = (x_46_re + -27.0) * (x_46_re * (x_46_re + x_46_im))
	else:
		tmp = -27.0 * (x_46_re * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.6e+221)
		tmp = Float64(Float64(x_46_re + -27.0) * Float64(x_46_re * Float64(x_46_re + x_46_im)));
	else
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3.6e+221)
		tmp = (x_46_re + -27.0) * (x_46_re * (x_46_re + x_46_im));
	else
		tmp = -27.0 * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 3.6e+221], N[(N[(x$46$re + -27.0), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.60000000000000009e221

   1. Initial program 83.9%

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

      1. difference-of-squares 85.6%

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

   3. Applied egg-rr 85.6%

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

   5. Simplified 51.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 36.1%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 36.1%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 36.1%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 38.6%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 38.6%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 38.6%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 38.6%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 45.7%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 45.7%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 49.5%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   if 3.60000000000000009e221 < x.im

   1. Initial program 75.4%

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

      1. difference-of-squares 87.9%

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

   3. Applied egg-rr 87.9%

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 20.7%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 20.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 20.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 20.7%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 20.7%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 20.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 20.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 20.7%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 20.7%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 27.0%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 27.0%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.re around 0 58.2%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

   11. Simplified 58.2%

       \[\leadsto \color{blue}{-27 \cdot \left(x.re \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.6 \cdot 10^{+221}:\\ \;\;\;\;\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 8: 31.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.28 \cdot 10^{+216}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.28e+216)
   (* x.im (* x.re (- x.re 27.0)))
   (* -27.0 (* x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.28e+216) {
		tmp = x_46_im * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= 1.28d+216) then
        tmp = x_46im * (x_46re * (x_46re - 27.0d0))
    else
        tmp = (-27.0d0) * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.28e+216) {
		tmp = x_46_im * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.28e+216:
		tmp = x_46_im * (x_46_re * (x_46_re - 27.0))
	else:
		tmp = -27.0 * (x_46_re * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.28e+216)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_re - 27.0)));
	else
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.28e+216)
		tmp = x_46_im * (x_46_re * (x_46_re - 27.0));
	else
		tmp = -27.0 * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.28e+216], N[(x$46$im * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.2799999999999999e216

   1. Initial program 84.2%

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

      1. difference-of-squares 85.9%

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

   3. Applied egg-rr 85.9%

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

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 36.3%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 36.3%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 36.3%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 38.8%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 38.8%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 38.8%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 38.8%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 45.9%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 45.9%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 49.7%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 49.7%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.im around inf 32.1%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

   if 1.2799999999999999e216 < x.im

   1. Initial program 71.3%

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

      1. difference-of-squares 83.1%

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

   3. Applied egg-rr 83.1%

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

   5. Simplified 49.4%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 19.7%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 19.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 19.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 19.7%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 19.7%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 19.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 19.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 19.7%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 19.7%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 25.6%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.re around 0 55.0%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

   11. Simplified 55.0%

       \[\leadsto \color{blue}{-27 \cdot \left(x.re \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.28 \cdot 10^{+216}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 9: 30.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.6 \cdot 10^{+214}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 3.6e+214) (* x.re (* x.re x.im)) (* -27.0 (* x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.6e+214) {
		tmp = x_46_re * (x_46_re * x_46_im);
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im 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 <= 3.6d+214) then
        tmp = x_46re * (x_46re * x_46im)
    else
        tmp = (-27.0d0) * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.6e+214) {
		tmp = x_46_re * (x_46_re * x_46_im);
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 3.6e+214:
		tmp = x_46_re * (x_46_re * x_46_im)
	else:
		tmp = -27.0 * (x_46_re * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.6e+214)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3.6e+214)
		tmp = x_46_re * (x_46_re * x_46_im);
	else
		tmp = -27.0 * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 3.6e+214], N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.6000000000000001e214

   1. Initial program 84.2%

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

      1. difference-of-squares 85.9%

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

   3. Applied egg-rr 85.9%

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

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 36.3%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 36.3%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 36.3%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 38.8%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 38.8%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 38.8%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 38.8%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 45.9%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 45.9%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 49.7%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 49.7%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.im around inf 32.1%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

   11. Simplified 27.8%

       \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(x.re + -27\right)\right)} \]

   12. Taylor expanded in x.re around inf 29.0%

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

   if 3.6000000000000001e214 < x.im

   1. Initial program 71.3%

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

      1. difference-of-squares 83.1%

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

   3. Applied egg-rr 83.1%

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

   5. Simplified 49.4%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 19.7%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   7. Step-by-step derivation

      1. sub-neg 19.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      2. metadata-eval 19.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      3. associate-*r* 19.7%

         \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      4. *-commutative 19.7%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

      5. sub-neg 19.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

      6. metadata-eval 19.7%

         \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

      7. distribute-rgt-out 19.7%

         \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

      8. unpow2 19.7%

         \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

      9. distribute-lft-in 25.6%

         \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

   9. Taylor expanded in x.re around 0 55.0%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

   11. Simplified 55.0%

       \[\leadsto \color{blue}{-27 \cdot \left(x.re \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.6 \cdot 10^{+214}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 10: 20.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ -27 \cdot \left(x.re \cdot x.im\right) \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* -27.0 (* x.re x.im)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return -27.0 * (x_46_re * x_46_im);
}

Fortran

NOTE: x.im 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 = (-27.0d0) * (x_46re * x_46im)
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return -27.0 * (x_46_re * x_46_im);
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return -27.0 * (x_46_re * x_46_im)

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(-27.0 * Float64(x_46_re * x_46_im))
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = -27.0 * (x_46_re * x_46_im);
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. difference-of-squares 85.7%

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

3. Applied egg-rr 85.7%

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

5. Simplified 51.2%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

6. Taylor expanded in x.im around 0 35.2%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
7. Step-by-step derivation

   1. sub-neg 35.2%

      \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

   2. metadata-eval 35.2%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

   3. associate-*r* 37.5%

      \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

   4. *-commutative 37.5%

      \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

   5. sub-neg 37.5%

      \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

   6. metadata-eval 37.5%

      \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

   7. distribute-rgt-out 44.1%

      \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

   8. unpow2 44.1%

      \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.re}\right) \]

   9. distribute-lft-in 48.1%

      \[\leadsto \left(x.re + -27\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.re\right)\right)} \]

8. Simplified 48.1%

   \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.im + x.re\right)\right)} \]

9. Taylor expanded in x.re around 0 24.9%

   \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

11. Simplified 24.9%

    \[\leadsto \color{blue}{-27 \cdot \left(x.re \cdot x.im\right)} \]

12. Final simplification 24.9%

    \[\leadsto -27 \cdot \left(x.re \cdot x.im\right) \]

Alternative 11: 3.1% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x.im 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
end function

Java

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

Python

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

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(-x_46_re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 81.5%

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

4. Taylor expanded in x.re around 0 53.5%

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

6. Simplified 3.4%

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

7. Final simplification 3.4%

   \[\leadsto -x.re \]

Developer target: 86.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023333 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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