math.cube on complex, real part

Percentage Accurate: 82.2% → 96.4%

Time: 10.1s

Alternatives: 12

Speedup: 0.9×

• 44.2% 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: 82.2% 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: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.im, x.im \cdot -3, {x.re\_m}^{3}\right)\\ \mathbf{elif}\;x.re\_m \leq 3.6 \cdot 10^{+170}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

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 2e+88)
    (fma (* x.re_m x.im) (* x.im -3.0) (pow x.re_m 3.0))
    (if (<= x.re_m 3.6e+170)
      (-
       (* 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 <= 2e+88) {
		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 <= 3.6e+170) {
		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 <= 2e+88)
		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 <= 3.6e+170)
		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, 2e+88], 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, 3.6e+170], 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 < 1.99999999999999992e88

   1. Initial program 87.1%

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

   3. Simplified 85.7%

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

      1. +-commutative 85.7%

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

      2. associate-*r* 93.1%

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

      3. fma-define 93.1%

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

   6. Applied egg-rr 93.1%

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

   if 1.99999999999999992e88 < x.re < 3.6e170

   1. Initial program 95.0%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 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 3.6e170 < x.re

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in x.re around inf 96.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{elif}\;x.re \leq 3.6 \cdot 10^{+170}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.5 \cdot 10^{+170}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

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 3.5e+170)
    (-
     (* 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 <= 3.5e+170) {
		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 <= 3.5d+170) 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 <= 3.5e+170) {
		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 <= 3.5e+170:
		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 <= 3.5e+170)
		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 <= 3.5e+170)
		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, 3.5e+170], 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 2 regimes

2. if x.re < 3.50000000000000005e170

   1. Initial program 87.7%

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

      1. difference-of-squares 90.8%

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

      2. *-commutative 90.8%

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

   4. Applied egg-rr 90.8%

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

   if 3.50000000000000005e170 < x.re

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in x.re around inf 96.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.5 \cdot 10^{+170}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.8 \cdot 10^{+170}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m + -27\right)\right)\\ \end{array} \end{array} \]

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 3.8e+170)
    (-
     (* 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))))
    (+ x.im (* x.re_m (* (+ x.re_m x.im) (+ x.re_m -27.0)))))))

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 <= 3.8e+170) {
		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_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	}
	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 <= 3.8d+170) 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_46im + (x_46re_m * ((x_46re_m + x_46im) * (x_46re_m + (-27.0d0))))
    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 <= 3.8e+170) {
		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_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	}
	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 <= 3.8e+170:
		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_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)))
	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 <= 3.8e+170)
		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 = Float64(x_46_im + Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m + -27.0))));
	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 <= 3.8e+170)
		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_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	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, 3.8e+170], 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[(x$46$im + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.7999999999999998e170

   1. Initial program 87.7%

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

      1. difference-of-squares 90.8%

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

      2. *-commutative 90.8%

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

   4. Applied egg-rr 90.8%

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

   if 3.7999999999999998e170 < x.re

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. distribute-lft-in 72.0%

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

      4. distribute-rgt-out 72.0%

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

      5. metadata-eval 72.0%

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

   4. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

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

      2. count-2 72.0%

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

      3. *-commutative 72.0%

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

      4. expm1-log1p-u 40.0%

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

      5. expm1-undefine 40.0%

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

      6. *-commutative 40.0%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 76.0%

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

      1. difference-of-squares 100.0%

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

   10. Applied egg-rr 100.0%

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

   12. Simplified 96.0%

       \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - -1 \cdot x.im \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.8 \cdot 10^{+170}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m + -27\right)\right)\\ \end{array} \end{array} \]

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 1.1e+154)
    (-
     (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (+ x.im (* x.re_m (* (+ x.re_m x.im) (+ x.re_m -27.0)))))))

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 <= 1.1e+154) {
		tmp = (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) * 2.0));
	} else {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	}
	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 <= 1.1d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = x_46im + (x_46re_m * ((x_46re_m + x_46im) * (x_46re_m + (-27.0d0))))
    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 <= 1.1e+154) {
		tmp = (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) * 2.0));
	} else {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	}
	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 <= 1.1e+154:
		tmp = (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) * 2.0))
	else:
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)))
	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 <= 1.1e+154)
		tmp = 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) * 2.0)));
	else
		tmp = Float64(x_46_im + Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m + -27.0))));
	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 <= 1.1e+154)
		tmp = (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) * 2.0));
	else
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	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, 1.1e+154], 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] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.1000000000000001e154

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. *-un-lft-identity 87.9%

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

      3. distribute-lft-in 87.9%

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

      4. distribute-rgt-out 87.9%

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

      5. metadata-eval 87.9%

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

   4. Applied egg-rr 87.9%

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

   if 1.1000000000000001e154 < x.re

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. distribute-lft-in 73.3%

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

      4. distribute-rgt-out 73.3%

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

      5. metadata-eval 73.3%

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

   4. Applied egg-rr 73.3%

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

      1. *-commutative 73.3%

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

      2. count-2 73.3%

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

      3. *-commutative 73.3%

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

      4. expm1-log1p-u 46.7%

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

      5. expm1-undefine 46.7%

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

      6. *-commutative 46.7%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 76.7%

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

      1. difference-of-squares 100.0%

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

   10. Applied egg-rr 100.0%

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

   12. Simplified 93.3%

       \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - -1 \cdot x.im \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;x.re\_m \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im + x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re\_m \cdot \left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m + -27\right)\right)\\ \end{array} \end{array} \]

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 3.2e-5)
    (* x.re_m (* -3.0 (* x.im x.im)))
    (if (<= x.re_m 1.35e+154)
      (+ x.im (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))))
      (+ x.im (* x.re_m (* (+ x.re_m x.im) (+ x.re_m -27.0))))))))

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 <= 3.2e-5) {
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)));
	} else {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	}
	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 <= 3.2d-5) then
        tmp = x_46re_m * ((-3.0d0) * (x_46im * x_46im))
    else if (x_46re_m <= 1.35d+154) then
        tmp = x_46im + (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im)))
    else
        tmp = x_46im + (x_46re_m * ((x_46re_m + x_46im) * (x_46re_m + (-27.0d0))))
    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 <= 3.2e-5) {
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)));
	} else {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	}
	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 <= 3.2e-5:
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im))
	elif x_46_re_m <= 1.35e+154:
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)))
	else:
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)))
	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 <= 3.2e-5)
		tmp = Float64(x_46_re_m * Float64(-3.0 * Float64(x_46_im * x_46_im)));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = Float64(x_46_im + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(x_46_im + Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m + -27.0))));
	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 <= 3.2e-5)
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)));
	else
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m + x_46_im) * (x_46_re_m + -27.0)));
	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, 3.2e-5], N[(x$46$re$95$m * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.35e+154], N[(x$46$im + 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]), $MachinePrecision], N[(x$46$im + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 3.19999999999999986e-5

   1. Initial program 85.4%

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

   3. Simplified 83.8%

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

      1. +-commutative 83.8%

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

      2. associate-*r* 92.3%

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

      3. fma-define 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in x.re around 0 66.8%

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

   9. Simplified 66.9%

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

       1. pow2 66.9%

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

   11. Applied egg-rr 66.9%

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

   if 3.19999999999999986e-5 < x.re < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. count-2 99.9%

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

      3. *-commutative 99.9%

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

      4. expm1-log1p-u 71.5%

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

      5. expm1-undefine 71.5%

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

      6. *-commutative 71.5%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 95.7%

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

      1. mul-1-neg 95.7%

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

      2. neg-sub0 95.7%

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

   10. Applied egg-rr 95.7%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(0 - x.im\right)} \]
   11. Step-by-step derivation

       1. neg-sub0 95.7%

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

   12. Simplified 95.7%

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

   if 1.35000000000000003e154 < x.re

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. distribute-lft-in 73.3%

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

      4. distribute-rgt-out 73.3%

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

      5. metadata-eval 73.3%

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

   4. Applied egg-rr 73.3%

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

      1. *-commutative 73.3%

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

      2. count-2 73.3%

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

      3. *-commutative 73.3%

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

      4. expm1-log1p-u 46.7%

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

      5. expm1-undefine 46.7%

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

      6. *-commutative 46.7%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 76.7%

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

      1. difference-of-squares 100.0%

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

   10. Applied egg-rr 100.0%

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

   12. Simplified 93.3%

       \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - -1 \cdot x.im \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im + x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;x.re\_m \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ \end{array} \end{array} \]

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 3.8e-5)
    (* x.re_m (* -3.0 (* x.im x.im)))
    (+ x.im (* x.re_m (- (* x.re_m x.re_m) (* 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) {
	double tmp;
	if (x_46_re_m <= 3.8e-5) {
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)));
	}
	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 <= 3.8d-5) then
        tmp = x_46re_m * ((-3.0d0) * (x_46im * x_46im))
    else
        tmp = x_46im + (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im)))
    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 <= 3.8e-5) {
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)));
	}
	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 <= 3.8e-5:
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im))
	else:
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)))
	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 <= 3.8e-5)
		tmp = Float64(x_46_re_m * Float64(-3.0 * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_im + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))));
	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 <= 3.8e-5)
		tmp = x_46_re_m * (-3.0 * (x_46_im * x_46_im));
	else
		tmp = x_46_im + (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)));
	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, 3.8e-5], N[(x$46$re$95$m * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im + 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]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.8000000000000002e-5

   1. Initial program 85.4%

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

   3. Simplified 83.8%

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

      1. +-commutative 83.8%

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

      2. associate-*r* 92.3%

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

      3. fma-define 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in x.re around 0 66.8%

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

   9. Simplified 66.9%

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

       1. pow2 66.9%

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

   11. Applied egg-rr 66.9%

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

   if 3.8000000000000002e-5 < x.re

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. *-un-lft-identity 88.3%

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

      3. distribute-lft-in 88.3%

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

      4. distribute-rgt-out 88.3%

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

      5. metadata-eval 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. count-2 88.3%

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

      3. *-commutative 88.3%

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

      4. expm1-log1p-u 60.7%

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

      5. expm1-undefine 60.7%

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

      6. *-commutative 60.7%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 87.5%

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

      1. mul-1-neg 87.5%

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

      2. neg-sub0 87.5%

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

   10. Applied egg-rr 87.5%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(0 - x.im\right)} \]
   11. Step-by-step derivation

       1. neg-sub0 87.5%

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

   12. Simplified 87.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.9% 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 86.2%

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

3. Simplified 83.9%

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

   1. +-commutative 83.9%

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

   2. associate-*r* 90.1%

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

   3. fma-define 90.1%

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

6. Applied egg-rr 90.1%

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

7. Taylor expanded in x.re around 0 56.4%

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

9. Simplified 56.5%

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

    1. pow2 56.5%

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

11. Applied egg-rr 56.5%

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

Alternative 8: 19.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot -27\right) \end{array} \]

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) -27.0)))

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) * -27.0);
}

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) * (-27.0d0))
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) * -27.0);
}

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) * -27.0)

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) * -27.0))
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) * -27.0);
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] * -27.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

   1. difference-of-squares 59.3%

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

4. Applied egg-rr 89.3%

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

6. Simplified 57.7%

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

7. Taylor expanded in x.re around 0 34.7%

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

8. Taylor expanded in x.im around 0 19.4%

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

9. Final simplification 19.4%

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

Alternative 9: 4.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot 27\right) \end{array} \]

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 27.0)))

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 * 27.0);
}

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 * 27.0d0)
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 * 27.0);
}

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 * 27.0)

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(x_46_re_m * 27.0))
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 * 27.0);
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[(x$46$re$95$m * 27.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

   1. difference-of-squares 59.3%

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

4. Applied egg-rr 89.3%

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

6. Simplified 57.7%

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

7. Taylor expanded in x.re around 0 34.7%

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

8. Taylor expanded in x.im around 0 19.4%

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

10. Simplified 4.7%

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

11. Taylor expanded in x.re around 0 4.7%

    \[\leadsto \color{blue}{27 \cdot x.re} \]
12. Step-by-step derivation

    1. *-commutative 4.7%

       \[\leadsto \color{blue}{x.re \cdot 27} \]

13. Simplified 4.7%

    \[\leadsto \color{blue}{x.re \cdot 27} \]
14. Add Preprocessing

Alternative 10: 3.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot -27\right) \end{array} \]

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 -27.0)))

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 * -27.0);
}

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 * (-27.0d0))
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 * -27.0);
}

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 * -27.0)

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(x_46_re_m * -27.0))
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 * -27.0);
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[(x$46$re$95$m * -27.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

   1. difference-of-squares 59.3%

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

4. Applied egg-rr 89.3%

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

6. Simplified 57.7%

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

7. Taylor expanded in x.re around 0 34.7%

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

8. Taylor expanded in x.im around 0 19.4%

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

10. Simplified 4.7%

    \[\leadsto \color{blue}{-27 \cdot \left(-x.re\right)} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 2.2%

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

    2. sqrt-unprod 19.5%

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

    3. sqr-neg 19.5%

       \[\leadsto -27 \cdot \sqrt{\color{blue}{x.re \cdot x.re}} \]

    4. sqrt-prod 1.6%

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

    5. add-sqr-sqrt 3.1%

       \[\leadsto -27 \cdot \color{blue}{x.re} \]

    6. pow1 3.1%

       \[\leadsto \color{blue}{{\left(-27 \cdot x.re\right)}^{1}} \]

    7. *-commutative 3.1%

       \[\leadsto {\color{blue}{\left(x.re \cdot -27\right)}}^{1} \]

12. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{{\left(x.re \cdot -27\right)}^{1}} \]
13. Step-by-step derivation

    1. unpow1 3.1%

       \[\leadsto \color{blue}{x.re \cdot -27} \]

14. Simplified 3.1%

    \[\leadsto \color{blue}{x.re \cdot -27} \]
15. Add Preprocessing

Alternative 11: 3.6% accurate, 19.0× speedup

Math

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

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.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_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_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_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_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 * 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_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 * x$46$im), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   2. *-un-lft-identity 86.2%

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

   3. distribute-lft-in 86.2%

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

   4. distribute-rgt-out 86.2%

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

   5. metadata-eval 86.2%

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

4. Applied egg-rr 86.2%

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

   1. *-commutative 86.2%

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

   2. count-2 86.2%

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

   3. *-commutative 86.2%

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

   4. expm1-log1p-u 66.4%

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

   5. expm1-undefine 54.4%

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

   6. *-commutative 54.4%

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

   7. flip-+ 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

6. Applied egg-rr 0.0%

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

8. Simplified 53.0%

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

9. Taylor expanded in x.re around 0 3.6%

   \[\leadsto \color{blue}{x.im} \]
10. Add Preprocessing

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot 8 \end{array} \]

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 8.0))

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 * 8.0;
}

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 * 8.0d0
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 * 8.0;
}

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 * 8.0

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 * 8.0)
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 * 8.0;
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 * 8.0), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

3. Simplified 83.9%

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

   1. flip-+ 21.0%

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

   2. unpow-prod-down 20.9%

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

   3. div-sub 20.9%

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

   4. pow2 20.9%

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

   5. pow-pow 21.0%

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

   6. metadata-eval 21.0%

      \[\leadsto \frac{{x.re}^{\color{blue}{6}}}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   7. *-commutative 21.0%

      \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{\left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re}} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   8. associate-*r* 21.0%

      \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot -3\right)} \cdot x.re} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   9. associate-*l* 21.0%

      \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-3 \cdot x.re\right)}} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   10. pow2 21.0%

       \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{{x.im}^{2}} \cdot \left(-3 \cdot x.re\right)} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

6. Applied egg-rr 15.6%

   \[\leadsto \color{blue}{\frac{{x.re}^{6}}{{x.re}^{3} - {x.im}^{2} \cdot \left(-3 \cdot x.re\right)} - \frac{{x.re}^{2} \cdot \left({x.im}^{4} \cdot 9\right)}{{x.re}^{3} - {x.im}^{2} \cdot \left(-3 \cdot x.re\right)}} \]

8. Simplified 2.8%

   \[\leadsto \color{blue}{8} \]
9. Add Preprocessing

Developer Target 1: 86.9% 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 2024177 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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