math.cube on complex, real part

Percentage Accurate: 82.6% → 96.3%

Time: 6.6s

Alternatives: 6

Speedup: 0.5×

• 45.1% 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: 82.6% 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: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.im, x.im \cdot -3, {x.re\_m}^{3}\right)\\ \mathbf{elif}\;x.re\_m \leq 9 \cdot 10^{+200}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5.5e+102)
    (fma (* x.re_m x.im) (* x.im -3.0) (pow x.re_m 3.0))
    (if (<= x.re_m 9e+200)
      (-
       (* x.re_m (* (+ x.re_m x.im) (- x.re_m x.im)))
       (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
      (pow x.re_m 3.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5.5e+102) {
		tmp = fma((x_46_re_m * x_46_im), (x_46_im * -3.0), pow(x_46_re_m, 3.0));
	} else if (x_46_re_m <= 9e+200) {
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5.5e+102)
		tmp = fma(Float64(x_46_re_m * x_46_im), Float64(x_46_im * -3.0), (x_46_re_m ^ 3.0));
	elseif (x_46_re_m <= 9e+200)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5.5e+102], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 9e+200], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 5.49999999999999981e102

   1. Initial program 83.8%

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

   3. Simplified 81.9%

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

      1. associate-*r* 81.9%

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

      2. associate-*l* 81.9%

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

      3. +-commutative 81.9%

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

      4. associate-*r* 89.0%

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

      5. associate-*r* 89.0%

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

      6. fma-define 90.4%

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

   6. Applied egg-rr 90.4%

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

   if 5.49999999999999981e102 < x.re < 8.99999999999999939e200

   1. Initial program 87.0%

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

      1. difference-of-squares 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 8.99999999999999939e200 < x.re

   1. Initial program 66.7%

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

      1. sqr-neg 66.7%

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

      2. *-commutative 66.7%

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

      3. fma-neg 66.7%

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

      4. sqr-neg 66.7%

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

      5. *-commutative 66.7%

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

      6. distribute-rgt-neg-in 66.7%

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

      7. *-commutative 66.7%

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

      8. count-2 66.7%

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

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf 95.2%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{elif}\;x.re \leq 9 \cdot 10^{+200}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -1 \cdot 10^{-320}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -1e-320)
    (* (* x.re_m x.im) (* x.im -3.0))
    (pow x.re_m 3.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-320) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-1d-320)) then
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    else
        tmp = x_46re_m ** 3.0d0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-320) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = Math.pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-320:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	else:
		tmp = math.pow(x_46_re_m, 3.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -1e-320)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-320)
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	else
		tmp = x_46_re_m ^ 3.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-320], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 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)) < -9.99989e-321

   1. Initial program 91.6%

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

      1. sqr-neg 91.6%

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

      2. *-commutative 91.6%

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

      3. fma-neg 91.6%

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

      4. sqr-neg 91.6%

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

      5. *-commutative 91.6%

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

      6. distribute-rgt-neg-in 91.6%

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

      7. *-commutative 91.6%

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

      8. count-2 91.6%

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

   3. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around 0 38.2%

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

   7. Simplified 38.2%

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

      1. unpow2 38.2%

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

      2. associate-*l* 38.2%

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

      3. associate-*l* 46.3%

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

      4. add-exp-log 25.1%

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

      5. add-exp-log 0.4%

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

      6. prod-exp 0.4%

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

   9. Applied egg-rr 0.4%

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

       1. exp-sum 0.4%

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

       2. rem-exp-log 19.0%

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

       3. *-commutative 19.0%

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

       4. rem-exp-log 46.3%

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

       5. *-commutative 46.3%

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

   11. Simplified 46.3%

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

   if -9.99989e-321 < (-.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 77.7%

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

      1. sqr-neg 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-neg 77.7%

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

      4. sqr-neg 77.7%

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

      5. *-commutative 77.7%

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

      6. distribute-rgt-neg-in 77.7%

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

      7. *-commutative 77.7%

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

      8. count-2 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf 64.9%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-320}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;{x.re\_m}^{3} + x.im \cdot \left(-3 \cdot \left(x.re\_m \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m - x.re\_m\right)\right)\\ \mathbf{elif}\;x.re\_m \leq 5.5 \cdot 10^{+200}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5.5e+102)
    (+
     (pow x.re_m 3.0)
     (* x.im (+ (* -3.0 (* x.re_m x.im)) (* x.re_m (- x.re_m x.re_m)))))
    (if (<= x.re_m 5.5e+200)
      (-
       (* x.re_m (* (+ x.re_m x.im) (- x.re_m x.im)))
       (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
      (pow x.re_m 3.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5.5e+102) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m))));
	} else if (x_46_re_m <= 5.5e+200) {
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 5.5d+102) then
        tmp = (x_46re_m ** 3.0d0) + (x_46im * (((-3.0d0) * (x_46re_m * x_46im)) + (x_46re_m * (x_46re_m - x_46re_m))))
    else if (x_46re_m <= 5.5d+200) then
        tmp = (x_46re_m * ((x_46re_m + x_46im) * (x_46re_m - x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))
    else
        tmp = x_46re_m ** 3.0d0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5.5e+102) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m))));
	} else if (x_46_re_m <= 5.5e+200) {
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = Math.pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 5.5e+102:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m))))
	elif x_46_re_m <= 5.5e+200:
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	else:
		tmp = math.pow(x_46_re_m, 3.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5.5e+102)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_im * Float64(Float64(-3.0 * Float64(x_46_re_m * x_46_im)) + Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)))));
	elseif (x_46_re_m <= 5.5e+200)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 5.5e+102)
		tmp = (x_46_re_m ^ 3.0) + (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m))));
	elseif (x_46_re_m <= 5.5e+200)
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5.5e+102], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$im * N[(N[(-3.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 5.5e+200], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 5.49999999999999981e102

   1. Initial program 83.8%

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

      1. difference-of-squares 85.2%

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in x.im around 0 89.0%

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

   6. Taylor expanded in x.re around 0 89.0%

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

   if 5.49999999999999981e102 < x.re < 5.5e200

   1. Initial program 87.0%

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

      1. difference-of-squares 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 5.5e200 < x.re

   1. Initial program 66.7%

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

      1. sqr-neg 66.7%

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

      2. *-commutative 66.7%

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

      3. fma-neg 66.7%

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

      4. sqr-neg 66.7%

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

      5. *-commutative 66.7%

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

      6. distribute-rgt-neg-in 66.7%

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

      7. *-commutative 66.7%

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

      8. count-2 66.7%

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

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf 95.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right) + x.re \cdot \left(x.re - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 5.5 \cdot 10^{+200}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - t\_0 \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right)\right) - t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))))
   (*
    x.re_s
    (if (<= (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) t_0) -5e-106)
      (* (* x.re_m x.im) (* x.im -3.0))
      (- (* x.re_m (* (+ x.re_m x.im) (- x.re_m x.im))) t_0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-106) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - t_0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im))
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - t_0) <= (-5d-106)) then
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    else
        tmp = (x_46re_m * ((x_46re_m + x_46im) * (x_46re_m - x_46im))) - t_0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-106) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - t_0;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im))
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-106:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	else:
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - t_0
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - t_0) <= -5e-106)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - x_46_im))) - t_0);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - t_0) <= -5e-106)
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	else
		tmp = (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im))) - t_0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], -5e-106], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$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)) < -4.99999999999999983e-106

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. *-commutative 90.3%

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

      3. fma-neg 90.3%

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

      4. sqr-neg 90.3%

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

      5. *-commutative 90.3%

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

      6. distribute-rgt-neg-in 90.3%

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

      7. *-commutative 90.3%

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

      8. count-2 90.3%

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

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around 0 35.2%

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

   7. Simplified 35.2%

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

      1. unpow2 35.2%

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

      2. associate-*l* 35.2%

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

      3. associate-*l* 44.8%

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

      4. add-exp-log 24.3%

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

      5. add-exp-log 0.3%

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

      6. prod-exp 0.3%

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

   9. Applied egg-rr 0.3%

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

       1. exp-sum 0.3%

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

       2. rem-exp-log 18.4%

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

       3. *-commutative 18.4%

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

       4. rem-exp-log 44.8%

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

       5. *-commutative 44.8%

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

   11. Simplified 44.8%

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

   if -4.99999999999999983e-106 < (-.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 79.3%

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

      1. difference-of-squares 83.2%

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

   4. Applied egg-rr 83.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* (* x.re_m x.im) (* x.im -3.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m * x_46im) * (x_46im * (-3.0d0)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0)))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. sqr-neg 82.7%

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

   2. *-commutative 82.7%

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

   3. fma-neg 82.7%

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

   4. sqr-neg 82.7%

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

   5. *-commutative 82.7%

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

   6. distribute-rgt-neg-in 82.7%

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

   7. *-commutative 82.7%

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

   8. count-2 82.7%

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

3. Simplified 82.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x.re around 0 45.4%

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

7. Simplified 45.4%

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

   1. unpow2 45.4%

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

   2. associate-*l* 45.4%

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

   3. associate-*l* 51.2%

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

   4. add-exp-log 27.5%

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

   5. add-exp-log 12.5%

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

   6. prod-exp 12.4%

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

9. Applied egg-rr 12.4%

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

    1. exp-sum 12.5%

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

    2. rem-exp-log 21.0%

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

    3. *-commutative 21.0%

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

    4. rem-exp-log 51.2%

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

    5. *-commutative 51.2%

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

11. Simplified 51.2%

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

12. Final simplification 51.2%

    \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right) \]
13. Add Preprocessing

Alternative 6: 50.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* x.re_m (* -3.0 (* x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * (-3.0 * (x_46_im * x_46_im)));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46re_m * ((-3.0d0) * (x_46im * x_46im)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * (-3.0 * (x_46_im * x_46_im)));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (x_46_re_m * (-3.0 * (x_46_im * x_46_im)))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_re_m * Float64(-3.0 * Float64(x_46_im * x_46_im))))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (x_46_re_m * (-3.0 * (x_46_im * x_46_im)));
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(x$46$re$95$m * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. sqr-neg 82.7%

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

   2. *-commutative 82.7%

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

   3. fma-neg 82.7%

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

   4. sqr-neg 82.7%

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

   5. *-commutative 82.7%

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

   6. distribute-rgt-neg-in 82.7%

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

   7. *-commutative 82.7%

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

   8. count-2 82.7%

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

3. Simplified 82.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x.re around 0 45.4%

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

7. Simplified 45.4%

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

   1. unpow2 45.4%

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

9. Applied egg-rr 45.4%

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

10. Final simplification 45.4%

    \[\leadsto x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right) \]
11. Add Preprocessing

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 2024106 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* 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)))