math.cos on complex, imaginary part

Percentage Accurate: 65.2% → 99.9%

Time: 4.2s

Alternatives: 10

Speedup: 1.3×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sinh \left(-im\right) \cdot \sin re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sinh (- im)) (sin re)))

C

double code(double re, double im) {
	return sinh(-im) * sin(re);
}

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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sinh(-im) * sin(re)
end function

Java

public static double code(double re, double im) {
	return Math.sinh(-im) * Math.sin(re);
}

Python

def code(re, im):
	return math.sinh(-im) * math.sin(re)

Julia

function code(re, im)
	return Float64(sinh(Float64(-im)) * sin(re))
end

MATLAB

function tmp = code(re, im)
	tmp = sinh(-im) * sin(re);
end

Wolfram

code[re_, im_] := N[(N[Sinh[(-im)], $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right) \cdot \frac{1}{2}} \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \frac{1}{2}\right)} \]

   6. lift--.f64 N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\left(e^{-im} - e^{im}\right)} \cdot \frac{1}{2}\right) \]

   7. sub-negate-rev N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)\right)} \cdot \frac{1}{2}\right) \]

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

      \[\leadsto \sin re \cdot \color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \frac{1}{2}\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   10. mult-flip N/A

       \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{e^{im} - e^{-im}}{2}}\right)\right) \]

   11. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{e^{im}} - e^{-im}}{2}\right)\right) \]

   12. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - \color{blue}{e^{-im}}}{2}\right)\right) \]

   13. lift-neg.f64 N/A

       \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2}\right)\right) \]

   14. sinh-def N/A

       \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \]

   15. sinh-neg N/A

       \[\leadsto \sin re \cdot \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \]

   16. lift-neg.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sinh \left(-im\right) \cdot \sin re} \]
4. Add Preprocessing

Alternative 2: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{re}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -2e-216)
     (/ 1.0 (* (/ -0.5 (sinh im)) (/ 2.0 re)))
     (if (<= t_0 5e-8)
       (* (sin re) (- im))
       (*
        (sinh (- im))
        (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -2e-216) {
		tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
	} else if (t_0 <= 5e-8) {
		tmp = sin(re) * -im;
	} else {
		tmp = sinh(-im) * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	}
	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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    if (t_0 <= (-2d-216)) then
        tmp = 1.0d0 / (((-0.5d0) / sinh(im)) * (2.0d0 / re))
    else if (t_0 <= 5d-8) then
        tmp = sin(re) * -im
    else
        tmp = sinh(-im) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (t_0 <= -2e-216) {
		tmp = 1.0 / ((-0.5 / Math.sinh(im)) * (2.0 / re));
	} else if (t_0 <= 5e-8) {
		tmp = Math.sin(re) * -im;
	} else {
		tmp = Math.sinh(-im) * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	tmp = 0
	if t_0 <= -2e-216:
		tmp = 1.0 / ((-0.5 / math.sinh(im)) * (2.0 / re))
	elif t_0 <= 5e-8:
		tmp = math.sin(re) * -im
	else:
		tmp = math.sinh(-im) * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -2e-216)
		tmp = Float64(1.0 / Float64(Float64(-0.5 / sinh(im)) * Float64(2.0 / re)));
	elseif (t_0 <= 5e-8)
		tmp = Float64(sin(re) * Float64(-im));
	else
		tmp = Float64(sinh(Float64(-im)) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	tmp = 0.0;
	if (t_0 <= -2e-216)
		tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
	elseif (t_0 <= 5e-8)
		tmp = sin(re) * -im;
	else
		tmp = sinh(-im) * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-216], N[(1.0 / N[(N[(-0.5 / N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(2.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-8], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[Sinh[(-im)], $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-216

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)} \]

      3. flip-- N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      10. div-flip N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}{e^{-im} + e^{im}}}}} \]

      11. flip-- N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      13. lower-/.f64 65.2

          \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{1}{e^{-im} - e^{im}}}} \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      15. sub-negate-rev N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{-2 \cdot \sinh im}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin re \cdot \frac{1}{2}}{\frac{1}{-2 \cdot \sinh im}}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      4. lower-/.f64 99.2

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot 0.5}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{\color{blue}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      9. metadata-eval 99.2

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{-0.5}}{\sinh im}}{\sin re \cdot 0.5}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{-0.5}{\sinh im}}{\sin re \cdot 0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{-1}{2}}{\sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

      4. lower-/.f64 99.2

         \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{1}{\sin re \cdot 0.5}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{\sinh im} \cdot \frac{1}{\sin re \cdot 0.5}}} \]

   8. Taylor expanded in re around 0

      \[\leadsto \frac{1}{\frac{\frac{-1}{2}}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 63.2

         \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{\color{blue}{re}}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]

   if -2.0000000000000001e-216 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 4.9999999999999998e-8

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{-im} - e^{im}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right) \cdot \frac{1}{2}} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \frac{1}{2}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(e^{-im} - e^{im}\right)} \cdot \frac{1}{2}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)\right)} \cdot \frac{1}{2}\right) \]

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

         \[\leadsto \sin re \cdot \color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \frac{1}{2}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      10. mult-flip N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{e^{im} - e^{-im}}{2}}\right)\right) \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{e^{im}} - e^{-im}}{2}\right)\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - \color{blue}{e^{-im}}}{2}\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2}\right)\right) \]

      14. sinh-def N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \]

      15. sinh-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \]

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

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

      1. lower-*.f64 52.3

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

   6. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.3

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(im\right)\right) \]

      6. lower-neg.f64 52.3

         \[\leadsto \sin re \cdot \left(-im\right) \]

   8. Applied rewrites 52.3%

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

   if 4.9999999999999998e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{-im} - e^{im}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right) \cdot \frac{1}{2}} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \frac{1}{2}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(e^{-im} - e^{im}\right)} \cdot \frac{1}{2}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)\right)} \cdot \frac{1}{2}\right) \]

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

         \[\leadsto \sin re \cdot \color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \frac{1}{2}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      10. mult-flip N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{e^{im} - e^{-im}}{2}}\right)\right) \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{e^{im}} - e^{-im}}{2}\right)\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - \color{blue}{e^{-im}}}{2}\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2}\right)\right) \]

      14. sinh-def N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \]

      15. sinh-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \]

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

      \[\leadsto \sinh \left(-im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {re}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{re}^{2}}\right)\right) \]

      4. lower-pow.f64 63.2

         \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{\color{blue}{2}}\right)\right) \]

   6. Applied rewrites 63.2%

      \[\leadsto \sinh \left(-im\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{re}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -2e-216)
     (/ 1.0 (* (/ -0.5 (sinh im)) (/ 2.0 re)))
     (if (<= t_0 5e-8)
       (* (sin re) (- im))
       (* (* -1.0 im) (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -2e-216) {
		tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
	} else if (t_0 <= 5e-8) {
		tmp = sin(re) * -im;
	} else {
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	}
	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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    if (t_0 <= (-2d-216)) then
        tmp = 1.0d0 / (((-0.5d0) / sinh(im)) * (2.0d0 / re))
    else if (t_0 <= 5d-8) then
        tmp = sin(re) * -im
    else
        tmp = ((-1.0d0) * im) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (t_0 <= -2e-216) {
		tmp = 1.0 / ((-0.5 / Math.sinh(im)) * (2.0 / re));
	} else if (t_0 <= 5e-8) {
		tmp = Math.sin(re) * -im;
	} else {
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	tmp = 0
	if t_0 <= -2e-216:
		tmp = 1.0 / ((-0.5 / math.sinh(im)) * (2.0 / re))
	elif t_0 <= 5e-8:
		tmp = math.sin(re) * -im
	else:
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -2e-216)
		tmp = Float64(1.0 / Float64(Float64(-0.5 / sinh(im)) * Float64(2.0 / re)));
	elseif (t_0 <= 5e-8)
		tmp = Float64(sin(re) * Float64(-im));
	else
		tmp = Float64(Float64(-1.0 * im) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	tmp = 0.0;
	if (t_0 <= -2e-216)
		tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
	elseif (t_0 <= 5e-8)
		tmp = sin(re) * -im;
	else
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-216], N[(1.0 / N[(N[(-0.5 / N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(2.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-8], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(-1.0 * im), $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-216

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)} \]

      3. flip-- N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      10. div-flip N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}{e^{-im} + e^{im}}}}} \]

      11. flip-- N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      13. lower-/.f64 65.2

          \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{1}{e^{-im} - e^{im}}}} \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      15. sub-negate-rev N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{-2 \cdot \sinh im}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin re \cdot \frac{1}{2}}{\frac{1}{-2 \cdot \sinh im}}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      4. lower-/.f64 99.2

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot 0.5}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{\color{blue}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      9. metadata-eval 99.2

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{-0.5}}{\sinh im}}{\sin re \cdot 0.5}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{-0.5}{\sinh im}}{\sin re \cdot 0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{-1}{2}}{\sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

      4. lower-/.f64 99.2

         \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{1}{\sin re \cdot 0.5}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{\sinh im} \cdot \frac{1}{\sin re \cdot 0.5}}} \]

   8. Taylor expanded in re around 0

      \[\leadsto \frac{1}{\frac{\frac{-1}{2}}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 63.2

         \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{\color{blue}{re}}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]

   if -2.0000000000000001e-216 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 4.9999999999999998e-8

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{-im} - e^{im}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right) \cdot \frac{1}{2}} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \frac{1}{2}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(e^{-im} - e^{im}\right)} \cdot \frac{1}{2}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)\right)} \cdot \frac{1}{2}\right) \]

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

         \[\leadsto \sin re \cdot \color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \frac{1}{2}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      10. mult-flip N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{e^{im} - e^{-im}}{2}}\right)\right) \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{e^{im}} - e^{-im}}{2}\right)\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - \color{blue}{e^{-im}}}{2}\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2}\right)\right) \]

      14. sinh-def N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \]

      15. sinh-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \]

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

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

      1. lower-*.f64 52.3

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

   6. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.3

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(im\right)\right) \]

      6. lower-neg.f64 52.3

         \[\leadsto \sin re \cdot \left(-im\right) \]

   8. Applied rewrites 52.3%

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

   if 4.9999999999999998e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{-im} - e^{im}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right) \cdot \frac{1}{2}} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \frac{1}{2}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(e^{-im} - e^{im}\right)} \cdot \frac{1}{2}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)\right)} \cdot \frac{1}{2}\right) \]

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

         \[\leadsto \sin re \cdot \color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \frac{1}{2}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      10. mult-flip N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{e^{im} - e^{-im}}{2}}\right)\right) \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{e^{im}} - e^{-im}}{2}\right)\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - \color{blue}{e^{-im}}}{2}\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2}\right)\right) \]

      14. sinh-def N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \]

      15. sinh-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \]

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

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

      1. lower-*.f64 52.3

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {re}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{re}^{2}}\right)\right) \]

      4. lower-pow.f64 36.8

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{\color{blue}{2}}\right)\right) \]

   9. Applied rewrites 36.8%

      \[\leadsto \left(-1 \cdot im\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{re}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -2e-216)
   (/ 1.0 (* (/ -0.5 (sinh im)) (/ 2.0 re)))
   (* (* -1.0 im) (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -2e-216) {
		tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
	} else {
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	}
	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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp(-im) - exp(im))) <= (-2d-216)) then
        tmp = 1.0d0 / (((-0.5d0) / sinh(im)) * (2.0d0 / re))
    else
        tmp = ((-1.0d0) * im) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im))) <= -2e-216) {
		tmp = 1.0 / ((-0.5 / Math.sinh(im)) * (2.0 / re));
	} else {
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))) <= -2e-216:
		tmp = 1.0 / ((-0.5 / math.sinh(im)) * (2.0 / re))
	else:
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -2e-216)
		tmp = Float64(1.0 / Float64(Float64(-0.5 / sinh(im)) * Float64(2.0 / re)));
	else
		tmp = Float64(Float64(-1.0 * im) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -2e-216)
		tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
	else
		tmp = (-1.0 * im) * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-216], N[(1.0 / N[(N[(-0.5 / N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(2.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * im), $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-216

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)} \]

      3. flip-- N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

      10. div-flip N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}{e^{-im} + e^{im}}}}} \]

      11. flip-- N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      13. lower-/.f64 65.2

          \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{1}{e^{-im} - e^{im}}}} \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

      15. sub-negate-rev N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{-2 \cdot \sinh im}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin re \cdot \frac{1}{2}}{\frac{1}{-2 \cdot \sinh im}}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      4. lower-/.f64 99.2

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot 0.5}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{\color{blue}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

      9. metadata-eval 99.2

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{-0.5}}{\sinh im}}{\sin re \cdot 0.5}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{-0.5}{\sinh im}}{\sin re \cdot 0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{-1}{2}}{\sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

      4. lower-/.f64 99.2

         \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{1}{\sin re \cdot 0.5}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{\sinh im} \cdot \frac{1}{\sin re \cdot 0.5}}} \]

   8. Taylor expanded in re around 0

      \[\leadsto \frac{1}{\frac{\frac{-1}{2}}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 63.2

         \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{\color{blue}{re}}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]

   if -2.0000000000000001e-216 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{-im} - e^{im}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(e^{-im} - e^{im}\right)\right) \cdot \frac{1}{2}} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \frac{1}{2}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(e^{-im} - e^{im}\right)} \cdot \frac{1}{2}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)\right)} \cdot \frac{1}{2}\right) \]

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

         \[\leadsto \sin re \cdot \color{blue}{\left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \frac{1}{2}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\left(e^{im} - e^{-im}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      10. mult-flip N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{e^{im} - e^{-im}}{2}}\right)\right) \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{e^{im}} - e^{-im}}{2}\right)\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - \color{blue}{e^{-im}}}{2}\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\frac{e^{im} - e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2}\right)\right) \]

      14. sinh-def N/A

          \[\leadsto \sin re \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \]

      15. sinh-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \]

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

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

      1. lower-*.f64 52.3

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {re}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{re}^{2}}\right)\right) \]

      4. lower-pow.f64 36.8

         \[\leadsto \left(-1 \cdot im\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{\color{blue}{2}}\right)\right) \]

   9. Applied rewrites 36.8%

      \[\leadsto \left(-1 \cdot im\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.0005:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -0.0005)
   (* (* 0.5 re) (- 1.0 (exp im)))
   (* (* re 0.5) (* (fma (* im im) -0.3333333333333333 -2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -0.0005) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = (re * 0.5) * (fma((im * im), -0.3333333333333333, -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -0.0005)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e-4

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.0%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   if -5.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 54.0

         \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   7. Applied rewrites 54.0%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.0

         \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.0

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      8. sub-flip N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

      10. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{3} + -2\right) \cdot im\right) \]

      12. lower-fma.f64 54.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left({im}^{2}, -0.3333333333333333, -2\right) \cdot im\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left({im}^{2}, \frac{-1}{3}, -2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]

      15. lower-*.f64 54.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

   9. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{-0.5}{\sinh im}}{0.5 \cdot re}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (/ (/ -0.5 (sinh im)) (* 0.5 re))))

C

double code(double re, double im) {
	return 1.0 / ((-0.5 / sinh(im)) / (0.5 * re));
}

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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (((-0.5d0) / sinh(im)) / (0.5d0 * re))
end function

Java

public static double code(double re, double im) {
	return 1.0 / ((-0.5 / Math.sinh(im)) / (0.5 * re));
}

Python

def code(re, im):
	return 1.0 / ((-0.5 / math.sinh(im)) / (0.5 * re))

Julia

function code(re, im)
	return Float64(1.0 / Float64(Float64(-0.5 / sinh(im)) / Float64(0.5 * re)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / ((-0.5 / sinh(im)) / (0.5 * re));
end

Wolfram

code[re_, im_] := N[(1.0 / N[(N[(-0.5 / N[Sinh[im], $MachinePrecision]), $MachinePrecision] / N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)} \]

   3. flip-- N/A

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

   4. div-flip N/A

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

   5. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

   10. div-flip N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}{e^{-im} + e^{im}}}}} \]

   11. flip-- N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

   12. lift--.f64 N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

   13. lower-/.f64 65.2

       \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{1}{e^{-im} - e^{im}}}} \]

   14. lift--.f64 N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

   15. sub-negate-rev N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)}}} \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{-2 \cdot \sinh im}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin re \cdot \frac{1}{2}}{\frac{1}{-2 \cdot \sinh im}}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

   4. lower-/.f64 99.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot 0.5}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\frac{1}{\color{blue}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   9. metadata-eval 99.2

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{-0.5}}{\sinh im}}{\sin re \cdot 0.5}} \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{-0.5}{\sinh im}}{\sin re \cdot 0.5}}} \]

6. Taylor expanded in re around 0

   \[\leadsto \frac{1}{\frac{\frac{\frac{-1}{2}}{\sinh im}}{\color{blue}{\frac{1}{2} \cdot re}}} \]
7. Step-by-step derivation

   1. lower-*.f64 63.2

      \[\leadsto \frac{1}{\frac{\frac{-0.5}{\sinh im}}{0.5 \cdot \color{blue}{re}}} \]

8. Applied rewrites 63.2%

   \[\leadsto \frac{1}{\frac{\frac{-0.5}{\sinh im}}{\color{blue}{0.5 \cdot re}}} \]
9. Add Preprocessing

Alternative 7: 49.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{re}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (* (/ -0.5 (sinh im)) (/ 2.0 re))))

C

double code(double re, double im) {
	return 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
}

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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (((-0.5d0) / sinh(im)) * (2.0d0 / re))
end function

Java

public static double code(double re, double im) {
	return 1.0 / ((-0.5 / Math.sinh(im)) * (2.0 / re));
}

Python

def code(re, im):
	return 1.0 / ((-0.5 / math.sinh(im)) * (2.0 / re))

Julia

function code(re, im)
	return Float64(1.0 / Float64(Float64(-0.5 / sinh(im)) * Float64(2.0 / re)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / ((-0.5 / sinh(im)) * (2.0 / re));
end

Wolfram

code[re_, im_] := N[(1.0 / N[(N[(-0.5 / N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(2.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)} \]

   3. flip-- N/A

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

   4. div-flip N/A

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

   5. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{-im} + e^{im}}{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}} \]

   10. div-flip N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}{e^{-im} + e^{im}}}}} \]

   11. flip-- N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

   12. lift--.f64 N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

   13. lower-/.f64 65.2

       \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{1}{e^{-im} - e^{im}}}} \]

   14. lift--.f64 N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{e^{-im} - e^{im}}}} \]

   15. sub-negate-rev N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\frac{1}{\color{blue}{\mathsf{neg}\left(\left(e^{im} - e^{-im}\right)\right)}}} \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{-2 \cdot \sinh im}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin re \cdot \frac{1}{2}}{\frac{1}{-2 \cdot \sinh im}}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

   4. lower-/.f64 99.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{-2 \cdot \sinh im}}{\sin re \cdot 0.5}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\frac{1}{\color{blue}{-2 \cdot \sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{-2}}{\sinh im}}}{\sin re \cdot \frac{1}{2}}} \]

   9. metadata-eval 99.2

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{-0.5}}{\sinh im}}{\sin re \cdot 0.5}} \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{-0.5}{\sinh im}}{\sin re \cdot 0.5}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{-1}{2}}{\sinh im}}{\sin re \cdot \frac{1}{2}}}} \]

   2. mult-flip N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2}}{\sinh im} \cdot \frac{1}{\sin re \cdot \frac{1}{2}}}} \]

   4. lower-/.f64 99.2

      \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{1}{\sin re \cdot 0.5}}} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{\sinh im} \cdot \frac{1}{\sin re \cdot 0.5}}} \]

8. Taylor expanded in re around 0

   \[\leadsto \frac{1}{\frac{\frac{-1}{2}}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]
9. Step-by-step derivation

   1. lower-/.f64 63.2

      \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \frac{2}{\color{blue}{re}}} \]

10. Applied rewrites 63.2%

    \[\leadsto \frac{1}{\frac{-0.5}{\sinh im} \cdot \color{blue}{\frac{2}{re}}} \]
11. Add Preprocessing

Alternative 8: 49.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* re 0.5) (* (fma (* im im) -0.3333333333333333 -2.0) im)))

C

double code(double re, double im) {
	return (re * 0.5) * (fma((im * im), -0.3333333333333333, -2.0) * im);
}

Julia

function code(re, im)
	return Float64(Float64(re * 0.5) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im))
end

Wolfram

code[re_, im_] := N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0

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

4. Applied rewrites 52.5%

   \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

5. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

   4. lower-pow.f64 54.0

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

7. Applied rewrites 54.0%

   \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 54.0

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 54.0

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

   7. lift--.f64 N/A

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

   8. sub-flip N/A

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

   10. *-commutative N/A

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{3} + -2\right) \cdot im\right) \]

   12. lower-fma.f64 54.0

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left({im}^{2}, -0.3333333333333333, -2\right) \cdot im\right) \]

   13. lift-pow.f64 N/A

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left({im}^{2}, \frac{-1}{3}, -2\right) \cdot im\right) \]

   14. unpow2 N/A

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]

   15. lower-*.f64 54.0

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

9. Applied rewrites 54.0%

   \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]
10. Add Preprocessing

Alternative 9: 43.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.0005:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(-2 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -0.0005)
   (* (* re 0.5) (* (* (* im im) im) -0.3333333333333333))
   (* (* 0.5 re) (* -2.0 im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -0.0005) {
		tmp = (re * 0.5) * (((im * im) * im) * -0.3333333333333333);
	} else {
		tmp = (0.5 * re) * (-2.0 * 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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp(-im) - exp(im))) <= (-0.0005d0)) then
        tmp = (re * 0.5d0) * (((im * im) * im) * (-0.3333333333333333d0))
    else
        tmp = (0.5d0 * re) * ((-2.0d0) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im))) <= -0.0005) {
		tmp = (re * 0.5) * (((im * im) * im) * -0.3333333333333333);
	} else {
		tmp = (0.5 * re) * (-2.0 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))) <= -0.0005:
		tmp = (re * 0.5) * (((im * im) * im) * -0.3333333333333333)
	else:
		tmp = (0.5 * re) * (-2.0 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -0.0005)
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(Float64(im * im) * im) * -0.3333333333333333));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(-2.0 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -0.0005)
		tmp = (re * 0.5) * (((im * im) * im) * -0.3333333333333333);
	else
		tmp = (0.5 * re) * (-2.0 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e-4

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 54.0

         \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   7. Applied rewrites 54.0%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]

   8. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 42.2

         \[\leadsto \left(0.5 \cdot re\right) \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

   10. Applied rewrites 42.2%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \left(-0.3333333333333333 \cdot \color{blue}{{im}^{3}}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 42.2

          \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{3} \cdot \frac{-1}{3}\right) \]

       6. lower-*.f64 42.2

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left({im}^{3} \cdot -0.3333333333333333\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{3} \cdot \frac{-1}{3}\right) \]

       8. unpow3 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot im\right) \cdot \frac{-1}{3}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot im\right) \cdot \frac{-1}{3}\right) \]

       11. unpow2 N/A

           \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \]

       12. lower-*.f64 42.2

           \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.3333333333333333\right) \]

   12. Applied rewrites 42.2%

       \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.3333333333333333\right)} \]

   if -5.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 33.9

         \[\leadsto \left(0.5 \cdot re\right) \cdot \left(-2 \cdot \color{blue}{im}\right) \]

   7. Applied rewrites 33.9%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 33.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(-2 \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) (* -2.0 im)))

C

double code(double re, double im) {
	return (0.5 * re) * (-2.0 * 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(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * ((-2.0d0) * im)
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * (-2.0 * im);
}

Python

def code(re, im):
	return (0.5 * re) * (-2.0 * im)

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(-2.0 * im))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * (-2.0 * im);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0

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

4. Applied rewrites 52.5%

   \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

5. Taylor expanded in im around 0

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

   1. lower-*.f64 33.9

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(-2 \cdot \color{blue}{im}\right) \]

7. Applied rewrites 33.9%

   \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))