math.cube on complex, real part

Percentage Accurate: 82.0% → 96.2%

Time: 14.3s

Alternatives: 5

Speedup: 1.4×

• 45.3% 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: 82.0% 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: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot x.im\_m\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 6.5e+146)
   (* (fma -3.0 (* x.im_m x.im_m) (* x.re x.re)) x.re)
   (* (* (* x.re x.im_m) x.im_m) -3.0)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 6.5e+146) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 6.5e+146)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * x_46_im_m) * -3.0);
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 6.5e+146], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.4999999999999997e146

   1. Initial program 85.2%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   if 6.4999999999999997e146 < x.im

   1. Initial program 57.1%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 94.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot x.im\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-308)
   (* (* (* -3.0 x.im_m) x.re) x.im_m)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-308)) then
        tmp = (((-3.0d0) * x_46im_m) * x_46re) * x_46im_m
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308:
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308)
		tmp = Float64(Float64(Float64(-3.0 * x_46_im_m) * x_46_re) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308)
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-308], N[(N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $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)) < -4.00000000000000013e-308

   1. Initial program 88.4%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 54.2%

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

   9. Applied rewrites 54.2%

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

   if -4.00000000000000013e-308 < (-.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. Add Preprocessing

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 66.7

         \[\leadsto \color{blue}{{x.re}^{3}} \]

   5. Applied rewrites 66.7%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot x.im\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-308)
   (* (* (* x.re x.im_m) x.im_m) -3.0)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308) {
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-308)) then
        tmp = ((x_46re * x_46im_m) * x_46im_m) * (-3.0d0)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308) {
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308:
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * x_46_im_m) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308)
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-308], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $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)) < -4.00000000000000013e-308

   1. Initial program 88.4%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 54.2%

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

   if -4.00000000000000013e-308 < (-.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. Add Preprocessing

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 66.7

         \[\leadsto \color{blue}{{x.re}^{3}} \]

   5. Applied rewrites 66.7%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\left(x.re \cdot \left(x.im\_m \cdot x.im\_m\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-308)
   (* (* x.re (* x.im_m x.im_m)) -3.0)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308) {
		tmp = (x_46_re * (x_46_im_m * x_46_im_m)) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-308)) then
        tmp = (x_46re * (x_46im_m * x_46im_m)) * (-3.0d0)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308) {
		tmp = (x_46_re * (x_46_im_m * x_46_im_m)) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308:
		tmp = (x_46_re * (x_46_im_m * x_46_im_m)) * -3.0
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im_m * x_46_im_m)) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-308)
		tmp = (x_46_re * (x_46_im_m * x_46_im_m)) * -3.0;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-308], N[(N[(x$46$re * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $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)) < -4.00000000000000013e-308

   1. Initial program 88.4%

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

   3. Taylor expanded in x.im around inf

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 42.8

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

   5. Applied rewrites 42.8%

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

   if -4.00000000000000013e-308 < (-.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. Add Preprocessing

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 66.7

         \[\leadsto \color{blue}{{x.re}^{3}} \]

   5. Applied rewrites 66.7%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.6% accurate, 3.6× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m) :precision binary64 (* (* x.re x.re) x.re))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return (x_46_re * x_46_re) * x_46_re;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = (x_46re * x_46re) * x_46re
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return (x_46_re * x_46_re) * x_46_re;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return (x_46_re * x_46_re) * x_46_re

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(Float64(x_46_re * x_46_re) * x_46_re)
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = (x_46_re * x_46_re) * x_46_re;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in x.im around 0

   \[\leadsto \color{blue}{{x.re}^{3}} \]
4. Step-by-step derivation

   1. lower-pow.f64 60.2

      \[\leadsto \color{blue}{{x.re}^{3}} \]

5. Applied rewrites 60.2%

   \[\leadsto \color{blue}{{x.re}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 60.2%

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

Developer Target 1: 87.0% 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 2024288 
(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)))