math.cube on complex, real part

Percentage Accurate: 82.8% → 96.6%

Time: 3.9s

Alternatives: 6

Speedup: 1.1×

• 55.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    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: 82.8% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    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: 96.6% accurate, 0.2× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 2 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(-3 \cdot \left|x.re\right|\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left|x.re\right| \leq 1.1 \cdot 10^{+198}:\\ \;\;\;\;\left|x.re\right| \cdot \left(\left(\left|x.re\right| - x.im\right) \cdot \left(x.im + \left|x.re\right|\right) - \left(x.im + x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left|x.re\right| \cdot \left(\left|x.re\right| + -1 \cdot \left|x.re\right|\right)\right) + {\left(\left|x.re\right|\right)}^{3}\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.re)
 (if (<= (fabs x.re) 2e-142)
   (* (* (* -3.0 (fabs x.re)) x.im) x.im)
   (if (<= (fabs x.re) 1.1e+198)
     (*
      (fabs x.re)
      (-
       (* (- (fabs x.re) x.im) (+ x.im (fabs x.re)))
       (* (+ x.im x.im) x.im)))
     (+
      (* x.im (* (fabs x.re) (+ (fabs x.re) (* -1.0 (fabs x.re)))))
      (pow (fabs x.re) 3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 2e-142) {
		tmp = ((-3.0 * fabs(x_46_re)) * x_46_im) * x_46_im;
	} else if (fabs(x_46_re) <= 1.1e+198) {
		tmp = fabs(x_46_re) * (((fabs(x_46_re) - x_46_im) * (x_46_im + fabs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im));
	} else {
		tmp = (x_46_im * (fabs(x_46_re) * (fabs(x_46_re) + (-1.0 * fabs(x_46_re))))) + pow(fabs(x_46_re), 3.0);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_re) <= 2e-142) {
		tmp = ((-3.0 * Math.abs(x_46_re)) * x_46_im) * x_46_im;
	} else if (Math.abs(x_46_re) <= 1.1e+198) {
		tmp = Math.abs(x_46_re) * (((Math.abs(x_46_re) - x_46_im) * (x_46_im + Math.abs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im));
	} else {
		tmp = (x_46_im * (Math.abs(x_46_re) * (Math.abs(x_46_re) + (-1.0 * Math.abs(x_46_re))))) + Math.pow(Math.abs(x_46_re), 3.0);
	}
	return Math.copySign(1.0, x_46_re) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_re) <= 2e-142:
		tmp = ((-3.0 * math.fabs(x_46_re)) * x_46_im) * x_46_im
	elif math.fabs(x_46_re) <= 1.1e+198:
		tmp = math.fabs(x_46_re) * (((math.fabs(x_46_re) - x_46_im) * (x_46_im + math.fabs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im))
	else:
		tmp = (x_46_im * (math.fabs(x_46_re) * (math.fabs(x_46_re) + (-1.0 * math.fabs(x_46_re))))) + math.pow(math.fabs(x_46_re), 3.0)
	return math.copysign(1.0, x_46_re) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 2e-142)
		tmp = Float64(Float64(Float64(-3.0 * abs(x_46_re)) * x_46_im) * x_46_im);
	elseif (abs(x_46_re) <= 1.1e+198)
		tmp = Float64(abs(x_46_re) * Float64(Float64(Float64(abs(x_46_re) - x_46_im) * Float64(x_46_im + abs(x_46_re))) - Float64(Float64(x_46_im + x_46_im) * x_46_im)));
	else
		tmp = Float64(Float64(x_46_im * Float64(abs(x_46_re) * Float64(abs(x_46_re) + Float64(-1.0 * abs(x_46_re))))) + (abs(x_46_re) ^ 3.0));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_re) <= 2e-142)
		tmp = ((-3.0 * abs(x_46_re)) * x_46_im) * x_46_im;
	elseif (abs(x_46_re) <= 1.1e+198)
		tmp = abs(x_46_re) * (((abs(x_46_re) - x_46_im) * (x_46_im + abs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im));
	else
		tmp = (x_46_im * (abs(x_46_re) * (abs(x_46_re) + (-1.0 * abs(x_46_re))))) + (abs(x_46_re) ^ 3.0);
	end
	tmp_2 = (sign(x_46_re) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 2e-142], N[(N[(N[(-3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 1.1e+198], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision] * N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] + N[(-1.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Abs[x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 2.0000000000000001e-142

   1. Initial program 82.8%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.0%

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 49.9%

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

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 56.0%

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

      23. lift-*.f64 N/A

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

      24. lift-neg.f64 N/A

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

      25. distribute-rgt-neg-out N/A

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

      26. distribute-lft-neg-in N/A

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

      27. metadata-eval N/A

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

      28. lower-*.f64 56.0%

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

   8. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 56.0%

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

   10. Applied rewrites 56.0%

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

   if 2.0000000000000001e-142 < x.re < 1.1e198

   1. Initial program 82.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

   if 1.1e198 < x.re

   1. Initial program 82.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in x.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 58.7%

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

   6. Applied rewrites 58.7%

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

Alternative 2: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 1.5 \cdot 10^{+152}:\\ \;\;\;\;\left(x.re \cdot x.re - \left(3 \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.im\right| \cdot x.re\right) \cdot \left(-3 \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (if (<= (fabs x.im) 1.5e+152)
  (* (- (* x.re x.re) (* (* 3.0 (fabs x.im)) (fabs x.im))) x.re)
  (* (* (fabs x.im) x.re) (* -3.0 (fabs x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 1.5e+152) {
		tmp = ((x_46_re * x_46_re) - ((3.0 * fabs(x_46_im)) * fabs(x_46_im))) * x_46_re;
	} else {
		tmp = (fabs(x_46_im) * x_46_re) * (-3.0 * fabs(x_46_im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (abs(x_46im) <= 1.5d+152) then
        tmp = ((x_46re * x_46re) - ((3.0d0 * abs(x_46im)) * abs(x_46im))) * x_46re
    else
        tmp = (abs(x_46im) * x_46re) * ((-3.0d0) * abs(x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_im) <= 1.5e+152) {
		tmp = ((x_46_re * x_46_re) - ((3.0 * Math.abs(x_46_im)) * Math.abs(x_46_im))) * x_46_re;
	} else {
		tmp = (Math.abs(x_46_im) * x_46_re) * (-3.0 * Math.abs(x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_im) <= 1.5e+152:
		tmp = ((x_46_re * x_46_re) - ((3.0 * math.fabs(x_46_im)) * math.fabs(x_46_im))) * x_46_re
	else:
		tmp = (math.fabs(x_46_im) * x_46_re) * (-3.0 * math.fabs(x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 1.5e+152)
		tmp = Float64(Float64(Float64(x_46_re * x_46_re) - Float64(Float64(3.0 * abs(x_46_im)) * abs(x_46_im))) * x_46_re);
	else
		tmp = Float64(Float64(abs(x_46_im) * x_46_re) * Float64(-3.0 * abs(x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_im) <= 1.5e+152)
		tmp = ((x_46_re * x_46_re) - ((3.0 * abs(x_46_im)) * abs(x_46_im))) * x_46_re;
	else
		tmp = (abs(x_46_im) * x_46_re) * (-3.0 * abs(x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 1.5e+152], N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[(3.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * N[(-3.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.5e152

   1. Initial program 82.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-out N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. associate--l+ N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-neg-out N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-*.f64 87.5%

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

   5. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.5%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. remove-double-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 87.5%

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

   7. Applied rewrites 87.5%

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

   if 1.5e152 < x.im

   1. Initial program 82.8%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.0%

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 49.9%

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

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 56.0%

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

      20. lift-*.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. distribute-rgt-neg-out N/A

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

      23. distribute-lft-neg-in N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 56.0%

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

   8. Applied rewrites 56.0%

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

Alternative 3: 56.0% accurate, 2.5× speedup

Math

\[\left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((-3.0d0) * x_46re) * x_46im) * x_46im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.0%

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

4. Applied rewrites 50.0%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 49.9%

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

6. Applied rewrites 49.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. distribute-lft-neg-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. associate-*r* N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 56.0%

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

   23. lift-*.f64 N/A

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

   24. lift-neg.f64 N/A

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

   25. distribute-rgt-neg-out N/A

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

   26. distribute-lft-neg-in N/A

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

   27. metadata-eval N/A

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

   28. lower-*.f64 56.0%

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

8. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 56.0%

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

10. Applied rewrites 56.0%

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

Alternative 4: 56.0% accurate, 2.5× speedup

Math

\[\left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((-3.0d0) * x_46im) * x_46re) * x_46im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.0%

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

4. Applied rewrites 50.0%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 49.9%

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

6. Applied rewrites 49.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. distribute-lft-neg-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. associate-*r* N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 56.0%

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

   23. lift-*.f64 N/A

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

   24. lift-neg.f64 N/A

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

   25. distribute-rgt-neg-out N/A

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

   26. distribute-lft-neg-in N/A

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

   27. metadata-eval N/A

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

   28. lower-*.f64 56.0%

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

8. Applied rewrites 56.0%

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

Alternative 5: 56.0% accurate, 2.5× speedup

Math

\[\left(x.im \cdot x.re\right) \cdot \left(-3 \cdot x.im\right) \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (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_im * x_46_re) * (-3.0 * x_46_im);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.0%

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

4. Applied rewrites 50.0%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 49.9%

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

6. Applied rewrites 49.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. distribute-lft-neg-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 56.0%

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

   20. lift-*.f64 N/A

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

   21. lift-neg.f64 N/A

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

   22. distribute-rgt-neg-out N/A

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

   23. distribute-lft-neg-in N/A

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

   24. metadata-eval N/A

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

   25. lower-*.f64 56.0%

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

8. Applied rewrites 56.0%

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

Alternative 6: 49.9% accurate, 2.5× speedup

Math

\[x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right) \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46re * ((-3.0d0) * (x_46im * x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.0%

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

4. Applied rewrites 50.0%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 49.9%

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

6. Applied rewrites 49.9%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64
  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))