math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

Alternatives: 13

Speedup: 1.5×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\cosh im \cdot \cos re \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (cosh im) (cos re)))

C

double code(double re, double im) {
	return cosh(im) * cos(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 = cosh(im) * cos(re)
end function

Java

public static double code(double re, double im) {
	return Math.cosh(im) * Math.cos(re);
}

Python

def code(re, im):
	return math.cosh(im) * math.cos(re)

Julia

function code(re, im)
	return Float64(cosh(im) * cos(re))
end

MATLAB

function tmp = code(re, im)
	tmp = cosh(im) * cos(re);
end

Wolfram

code[re_, im_] := N[(N[Cosh[im], $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. mult-flip-rev N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. cosh-def N/A

       \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   14. lower-cosh.f64 100.0%

       \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]
4. Add Preprocessing

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ t_2 := 1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\\ \mathbf{if}\;t\_1 \leq -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot t\_2\right)\\ \mathbf{elif}\;t\_1 \leq \frac{9007199254737117}{9007199254740992}:\\ \;\;\;\;t\_0 \cdot \left(\left(2 + im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 1/2 (cos re)))
       (t_1 (* t_0 (+ (exp (- im)) (exp im))))
       (t_2 (+ 1 (* im (+ 1 (* 1/2 im))))))
  (if (<=
       t_1
       -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704)
    (*
     (+ 1/2 (* -1/4 (pow re 2)))
     (* (+ 2 (* im (- (* 2 im) 2))) t_2))
    (if (<= t_1 9007199254737117/9007199254740992)
      (* t_0 (* (+ 2 (* im (- (* im (+ 2 (* -4/3 im))) 2))) t_2))
      (* (cosh im) 1)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double t_2 = 1.0 + (im * (1.0 + (0.5 * im)));
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = (0.5 + (-0.25 * pow(re, 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * t_2);
	} else if (t_1 <= 0.9999999999995698) {
		tmp = t_0 * ((2.0 + (im * ((im * (2.0 + (-1.3333333333333333 * im))) - 2.0))) * t_2);
	} else {
		tmp = cosh(im) * 1.0;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = t_0 * (exp(-im) + exp(im))
    t_2 = 1.0d0 + (im * (1.0d0 + (0.5d0 * im)))
    if (t_1 <= (-5d+172)) then
        tmp = (0.5d0 + ((-0.25d0) * (re ** 2.0d0))) * ((2.0d0 + (im * ((2.0d0 * im) - 2.0d0))) * t_2)
    else if (t_1 <= 0.9999999999995698d0) then
        tmp = t_0 * ((2.0d0 + (im * ((im * (2.0d0 + ((-1.3333333333333333d0) * im))) - 2.0d0))) * t_2)
    else
        tmp = cosh(im) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = t_0 * (Math.exp(-im) + Math.exp(im));
	double t_2 = 1.0 + (im * (1.0 + (0.5 * im)));
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = (0.5 + (-0.25 * Math.pow(re, 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * t_2);
	} else if (t_1 <= 0.9999999999995698) {
		tmp = t_0 * ((2.0 + (im * ((im * (2.0 + (-1.3333333333333333 * im))) - 2.0))) * t_2);
	} else {
		tmp = Math.cosh(im) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = t_0 * (math.exp(-im) + math.exp(im))
	t_2 = 1.0 + (im * (1.0 + (0.5 * im)))
	tmp = 0
	if t_1 <= -5e+172:
		tmp = (0.5 + (-0.25 * math.pow(re, 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * t_2)
	elif t_1 <= 0.9999999999995698:
		tmp = t_0 * ((2.0 + (im * ((im * (2.0 + (-1.3333333333333333 * im))) - 2.0))) * t_2)
	else:
		tmp = math.cosh(im) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	t_2 = Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))
	tmp = 0.0
	if (t_1 <= -5e+172)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * (re ^ 2.0))) * Float64(Float64(2.0 + Float64(im * Float64(Float64(2.0 * im) - 2.0))) * t_2));
	elseif (t_1 <= 0.9999999999995698)
		tmp = Float64(t_0 * Float64(Float64(2.0 + Float64(im * Float64(Float64(im * Float64(2.0 + Float64(-1.3333333333333333 * im))) - 2.0))) * t_2));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = t_0 * (exp(-im) + exp(im));
	t_2 = 1.0 + (im * (1.0 + (0.5 * im)));
	tmp = 0.0;
	if (t_1 <= -5e+172)
		tmp = (0.5 + (-0.25 * (re ^ 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * t_2);
	elseif (t_1 <= 0.9999999999995698)
		tmp = t_0 * ((2.0 + (im * ((im * (2.0 + (-1.3333333333333333 * im))) - 2.0))) * t_2);
	else
		tmp = cosh(im) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1 + N[(im * N[(1 + N[(1/2 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704], N[(N[(1/2 + N[(-1/4 * N[Power[re, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2 + N[(im * N[(N[(2 * im), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 9007199254737117/9007199254740992], N[(t$95$0 * N[(N[(2 + N[(im * N[(N[(im * N[(2 + N[(-4/3 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e172

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

   10. Taylor expanded in re around 0

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

       1. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       3. lower-pow.f64 55.9%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

   12. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

   if -5.0000000000000001e172 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999956979

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

   10. Taylor expanded in im around 0

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

       1. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(2 + \color{blue}{im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)}\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(2 + im \cdot \color{blue}{\left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)}\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       3. lower--.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(2 + im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - \color{blue}{2}\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(2 + im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(2 + im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       6. lower-*.f64 69.7%

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(2 + im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

   12. Applied rewrites 69.7%

       \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(2 + im \cdot \left(im \cdot \left(2 + \frac{-4}{3} \cdot im\right) - 2\right)\right)} \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

   if 0.99999999999956979 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq \frac{9007199254737117}{9007199254740992}:\\ \;\;\;\;\left(\left(\cos re \cdot \left(\left(im \cdot \frac{1}{2} - -1\right) \cdot im - -1\right)\right) \cdot \left(\left(\left(im + im\right) - 2\right) \cdot im - -2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im)))))
  (if (<=
       t_0
       -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704)
    (*
     (+ 1/2 (* -1/4 (pow re 2)))
     (* (+ 2 (* im (- (* 2 im) 2))) (+ 1 (* im (+ 1 (* 1/2 im))))))
    (if (<= t_0 9007199254737117/9007199254740992)
      (*
       (*
        (* (cos re) (- (* (- (* im 1/2) -1) im) -1))
        (- (* (- (+ im im) 2) im) -2))
       1/2)
      (* (cosh im) 1)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -5e+172) {
		tmp = (0.5 + (-0.25 * pow(re, 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (t_0 <= 0.9999999999995698) {
		tmp = ((cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5;
	} else {
		tmp = cosh(im) * 1.0;
	}
	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 * cos(re)) * (exp(-im) + exp(im))
    if (t_0 <= (-5d+172)) then
        tmp = (0.5d0 + ((-0.25d0) * (re ** 2.0d0))) * ((2.0d0 + (im * ((2.0d0 * im) - 2.0d0))) * (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (t_0 <= 0.9999999999995698d0) then
        tmp = ((cos(re) * ((((im * 0.5d0) - (-1.0d0)) * im) - (-1.0d0))) * ((((im + im) - 2.0d0) * im) - (-2.0d0))) * 0.5d0
    else
        tmp = cosh(im) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -5e+172) {
		tmp = (0.5 + (-0.25 * Math.pow(re, 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (t_0 <= 0.9999999999995698) {
		tmp = ((Math.cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5;
	} else {
		tmp = Math.cosh(im) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -5e+172:
		tmp = (0.5 + (-0.25 * math.pow(re, 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * (1.0 + (im * (1.0 + (0.5 * im)))))
	elif t_0 <= 0.9999999999995698:
		tmp = ((math.cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5
	else:
		tmp = math.cosh(im) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -5e+172)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * (re ^ 2.0))) * Float64(Float64(2.0 + Float64(im * Float64(Float64(2.0 * im) - 2.0))) * Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (t_0 <= 0.9999999999995698)
		tmp = Float64(Float64(Float64(cos(re) * Float64(Float64(Float64(Float64(im * 0.5) - -1.0) * im) - -1.0)) * Float64(Float64(Float64(Float64(im + im) - 2.0) * im) - -2.0)) * 0.5);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -5e+172)
		tmp = (0.5 + (-0.25 * (re ^ 2.0))) * ((2.0 + (im * ((2.0 * im) - 2.0))) * (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (t_0 <= 0.9999999999995698)
		tmp = ((cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5;
	else
		tmp = cosh(im) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704], N[(N[(1/2 + N[(-1/4 * N[Power[re, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2 + N[(im * N[(N[(2 * im), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1 + N[(im * N[(1 + N[(1/2 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 9007199254737117/9007199254740992], N[(N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im * 1/2), $MachinePrecision] - -1), $MachinePrecision] * im), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(im + im), $MachinePrecision] - 2), $MachinePrecision] * im), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e172

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

   10. Taylor expanded in re around 0

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

       1. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

       3. lower-pow.f64 55.9%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

   12. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(2 + im \cdot \left(2 \cdot im - 2\right)\right) \cdot \left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)\right) \]

   if -5.0000000000000001e172 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999956979

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 87.6%

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

   if 0.99999999999956979 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \left(\left(re \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{if}\;t\_0 \leq -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}\right) \cdot 2\\ \mathbf{elif}\;t\_0 \leq \frac{9007199254737117}{9007199254740992}:\\ \;\;\;\;\left(\left(\cos re \cdot \left(\left(im \cdot \frac{1}{2} - -1\right) \cdot im - -1\right)\right) \cdot \left(\left(\left(im + im\right) - 2\right) \cdot im - -2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im))))
       (t_1 (* (* (* re re) re) re)))
  (if (<=
       t_0
       -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704)
    (* (+ 1/2 (* -1/4 (sqrt (sqrt (* t_1 t_1))))) 2)
    (if (<= t_0 9007199254737117/9007199254740992)
      (*
       (*
        (* (cos re) (- (* (- (* im 1/2) -1) im) -1))
        (- (* (- (+ im im) 2) im) -2))
       1/2)
      (* (cosh im) 1)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double t_1 = ((re * re) * re) * re;
	double tmp;
	if (t_0 <= -5e+172) {
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_1 * t_1))))) * 2.0;
	} else if (t_0 <= 0.9999999999995698) {
		tmp = ((cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5;
	} else {
		tmp = cosh(im) * 1.0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
    t_1 = ((re * re) * re) * re
    if (t_0 <= (-5d+172)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(sqrt((t_1 * t_1))))) * 2.0d0
    else if (t_0 <= 0.9999999999995698d0) then
        tmp = ((cos(re) * ((((im * 0.5d0) - (-1.0d0)) * im) - (-1.0d0))) * ((((im + im) - 2.0d0) * im) - (-2.0d0))) * 0.5d0
    else
        tmp = cosh(im) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double t_1 = ((re * re) * re) * re;
	double tmp;
	if (t_0 <= -5e+172) {
		tmp = (0.5 + (-0.25 * Math.sqrt(Math.sqrt((t_1 * t_1))))) * 2.0;
	} else if (t_0 <= 0.9999999999995698) {
		tmp = ((Math.cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5;
	} else {
		tmp = Math.cosh(im) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	t_1 = ((re * re) * re) * re
	tmp = 0
	if t_0 <= -5e+172:
		tmp = (0.5 + (-0.25 * math.sqrt(math.sqrt((t_1 * t_1))))) * 2.0
	elif t_0 <= 0.9999999999995698:
		tmp = ((math.cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5
	else:
		tmp = math.cosh(im) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = Float64(Float64(Float64(re * re) * re) * re)
	tmp = 0.0
	if (t_0 <= -5e+172)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(sqrt(Float64(t_1 * t_1))))) * 2.0);
	elseif (t_0 <= 0.9999999999995698)
		tmp = Float64(Float64(Float64(cos(re) * Float64(Float64(Float64(Float64(im * 0.5) - -1.0) * im) - -1.0)) * Float64(Float64(Float64(Float64(im + im) - 2.0) * im) - -2.0)) * 0.5);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	t_1 = ((re * re) * re) * re;
	tmp = 0.0;
	if (t_0 <= -5e+172)
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_1 * t_1))))) * 2.0;
	elseif (t_0 <= 0.9999999999995698)
		tmp = ((cos(re) * ((((im * 0.5) - -1.0) * im) - -1.0)) * ((((im + im) - 2.0) * im) - -2.0)) * 0.5;
	else
		tmp = cosh(im) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[t$95$0, 9007199254737117/9007199254740992], N[(N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im * 1/2), $MachinePrecision] - -1), $MachinePrecision] * im), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(im + im), $MachinePrecision] - 2), $MachinePrecision] * im), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e172

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}\right) \cdot 2 \]

       2. sqrt-unprod N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       4. lower-*.f64 37.0%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       7. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       11. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       14. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)}}\right) \cdot 2 \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       17. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

       18. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   11. Applied rewrites 37.0%

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   if -5.0000000000000001e172 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999956979

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 87.6%

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

   if 0.99999999999956979 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-\left|im\right|} + e^{\left|im\right|}\right)\\ t_2 := \left(\left(re \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{if}\;t\_1 \leq -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{t\_2 \cdot t\_2}}\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq \frac{9007199254737117}{9007199254740992}:\\ \;\;\;\;t\_0 \cdot \left(\left(2 + \left|im\right| \cdot \left(2 \cdot \left|im\right| - 2\right)\right) \cdot \left(1 + \left|im\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh \left(\left|im\right|\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 1/2 (cos re)))
       (t_1 (* t_0 (+ (exp (- (fabs im))) (exp (fabs im)))))
       (t_2 (* (* (* re re) re) re)))
  (if (<=
       t_1
       -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704)
    (* (+ 1/2 (* -1/4 (sqrt (sqrt (* t_2 t_2))))) 2)
    (if (<= t_1 9007199254737117/9007199254740992)
      (*
       t_0
       (* (+ 2 (* (fabs im) (- (* 2 (fabs im)) 2))) (+ 1 (fabs im))))
      (* (cosh (fabs im)) 1)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-fabs(im)) + exp(fabs(im)));
	double t_2 = ((re * re) * re) * re;
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_2 * t_2))))) * 2.0;
	} else if (t_1 <= 0.9999999999995698) {
		tmp = t_0 * ((2.0 + (fabs(im) * ((2.0 * fabs(im)) - 2.0))) * (1.0 + fabs(im)));
	} else {
		tmp = cosh(fabs(im)) * 1.0;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = t_0 * (exp(-abs(im)) + exp(abs(im)))
    t_2 = ((re * re) * re) * re
    if (t_1 <= (-5d+172)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(sqrt((t_2 * t_2))))) * 2.0d0
    else if (t_1 <= 0.9999999999995698d0) then
        tmp = t_0 * ((2.0d0 + (abs(im) * ((2.0d0 * abs(im)) - 2.0d0))) * (1.0d0 + abs(im)))
    else
        tmp = cosh(abs(im)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = t_0 * (Math.exp(-Math.abs(im)) + Math.exp(Math.abs(im)));
	double t_2 = ((re * re) * re) * re;
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = (0.5 + (-0.25 * Math.sqrt(Math.sqrt((t_2 * t_2))))) * 2.0;
	} else if (t_1 <= 0.9999999999995698) {
		tmp = t_0 * ((2.0 + (Math.abs(im) * ((2.0 * Math.abs(im)) - 2.0))) * (1.0 + Math.abs(im)));
	} else {
		tmp = Math.cosh(Math.abs(im)) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = t_0 * (math.exp(-math.fabs(im)) + math.exp(math.fabs(im)))
	t_2 = ((re * re) * re) * re
	tmp = 0
	if t_1 <= -5e+172:
		tmp = (0.5 + (-0.25 * math.sqrt(math.sqrt((t_2 * t_2))))) * 2.0
	elif t_1 <= 0.9999999999995698:
		tmp = t_0 * ((2.0 + (math.fabs(im) * ((2.0 * math.fabs(im)) - 2.0))) * (1.0 + math.fabs(im)))
	else:
		tmp = math.cosh(math.fabs(im)) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-abs(im))) + exp(abs(im))))
	t_2 = Float64(Float64(Float64(re * re) * re) * re)
	tmp = 0.0
	if (t_1 <= -5e+172)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(sqrt(Float64(t_2 * t_2))))) * 2.0);
	elseif (t_1 <= 0.9999999999995698)
		tmp = Float64(t_0 * Float64(Float64(2.0 + Float64(abs(im) * Float64(Float64(2.0 * abs(im)) - 2.0))) * Float64(1.0 + abs(im))));
	else
		tmp = Float64(cosh(abs(im)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = t_0 * (exp(-abs(im)) + exp(abs(im)));
	t_2 = ((re * re) * re) * re;
	tmp = 0.0;
	if (t_1 <= -5e+172)
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_2 * t_2))))) * 2.0;
	elseif (t_1 <= 0.9999999999995698)
		tmp = t_0 * ((2.0 + (abs(im) * ((2.0 * abs(im)) - 2.0))) * (1.0 + abs(im)));
	else
		tmp = cosh(abs(im)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] + N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[t$95$1, 9007199254737117/9007199254740992], N[(t$95$0 * N[(N[(2 + N[(N[Abs[im], $MachinePrecision] * N[(N[(2 * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1 + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[N[Abs[im], $MachinePrecision]], $MachinePrecision] * 1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e172

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}\right) \cdot 2 \]

       2. sqrt-unprod N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       4. lower-*.f64 37.0%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       7. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       11. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       14. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)}}\right) \cdot 2 \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       17. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

       18. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   11. Applied rewrites 37.0%

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   if -5.0000000000000001e172 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999956979

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 66.7%

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

   9. Applied rewrites 66.7%

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

   if 0.99999999999956979 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ t_2 := \left(\left(re \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{if}\;t\_1 \leq -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{t\_2 \cdot t\_2}}\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq \frac{9007199254737117}{9007199254740992}:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 1/2 (cos re)))
       (t_1 (* t_0 (+ (exp (- im)) (exp im))))
       (t_2 (* (* (* re re) re) re)))
  (if (<=
       t_1
       -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704)
    (* (+ 1/2 (* -1/4 (sqrt (sqrt (* t_2 t_2))))) 2)
    (if (<= t_1 9007199254737117/9007199254740992)
      (* t_0 2)
      (* (cosh im) 1)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double t_2 = ((re * re) * re) * re;
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_2 * t_2))))) * 2.0;
	} else if (t_1 <= 0.9999999999995698) {
		tmp = t_0 * 2.0;
	} else {
		tmp = cosh(im) * 1.0;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = t_0 * (exp(-im) + exp(im))
    t_2 = ((re * re) * re) * re
    if (t_1 <= (-5d+172)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(sqrt((t_2 * t_2))))) * 2.0d0
    else if (t_1 <= 0.9999999999995698d0) then
        tmp = t_0 * 2.0d0
    else
        tmp = cosh(im) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = t_0 * (Math.exp(-im) + Math.exp(im));
	double t_2 = ((re * re) * re) * re;
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = (0.5 + (-0.25 * Math.sqrt(Math.sqrt((t_2 * t_2))))) * 2.0;
	} else if (t_1 <= 0.9999999999995698) {
		tmp = t_0 * 2.0;
	} else {
		tmp = Math.cosh(im) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = t_0 * (math.exp(-im) + math.exp(im))
	t_2 = ((re * re) * re) * re
	tmp = 0
	if t_1 <= -5e+172:
		tmp = (0.5 + (-0.25 * math.sqrt(math.sqrt((t_2 * t_2))))) * 2.0
	elif t_1 <= 0.9999999999995698:
		tmp = t_0 * 2.0
	else:
		tmp = math.cosh(im) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	t_2 = Float64(Float64(Float64(re * re) * re) * re)
	tmp = 0.0
	if (t_1 <= -5e+172)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(sqrt(Float64(t_2 * t_2))))) * 2.0);
	elseif (t_1 <= 0.9999999999995698)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = t_0 * (exp(-im) + exp(im));
	t_2 = ((re * re) * re) * re;
	tmp = 0.0;
	if (t_1 <= -5e+172)
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_2 * t_2))))) * 2.0;
	elseif (t_1 <= 0.9999999999995698)
		tmp = t_0 * 2.0;
	else
		tmp = cosh(im) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000000701959312789985260891230985285064568046915021472510652274325054054092066621782843422306142881889050953096494638431569844936383886042210844858380302841544704], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[t$95$1, 9007199254737117/9007199254740992], N[(t$95$0 * 2), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.0000000000000001e172

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}\right) \cdot 2 \]

       2. sqrt-unprod N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       4. lower-*.f64 37.0%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       7. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       11. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       14. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)}}\right) \cdot 2 \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       17. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

       18. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   11. Applied rewrites 37.0%

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   if -5.0000000000000001e172 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999956979

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   if 0.99999999999956979 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(re \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{if}\;\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (* (* re re) re) re)))
  (if (<=
       (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im)))
       -3602879701896397/72057594037927936)
    (* (+ 1/2 (* -1/4 (sqrt (sqrt (* t_0 t_0))))) 2)
    (* (cosh im) 1))))

C

double code(double re, double im) {
	double t_0 = ((re * re) * re) * re;
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_0 * t_0))))) * 2.0;
	} else {
		tmp = cosh(im) * 1.0;
	}
	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 = ((re * re) * re) * re
    if (((0.5d0 * cos(re)) * (exp(-im) + exp(im))) <= (-0.05d0)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(sqrt((t_0 * t_0))))) * 2.0d0
    else
        tmp = cosh(im) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((re * re) * re) * re;
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im))) <= -0.05) {
		tmp = (0.5 + (-0.25 * Math.sqrt(Math.sqrt((t_0 * t_0))))) * 2.0;
	} else {
		tmp = Math.cosh(im) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((re * re) * re) * re
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= -0.05:
		tmp = (0.5 + (-0.25 * math.sqrt(math.sqrt((t_0 * t_0))))) * 2.0
	else:
		tmp = math.cosh(im) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(re * re) * re) * re)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(sqrt(Float64(t_0 * t_0))))) * 2.0);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((re * re) * re) * re;
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05)
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_0 * t_0))))) * 2.0;
	else
		tmp = cosh(im) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3602879701896397/72057594037927936], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}\right) \cdot 2 \]

       2. sqrt-unprod N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       4. lower-*.f64 37.0%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       7. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       11. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       14. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)}}\right) \cdot 2 \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       17. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

       18. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   11. Applied rewrites 37.0%

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(re \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{if}\;\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-\left|im\right|} + e^{\left|im\right|}\right) \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(2 + \left|im\right| \cdot \left(2 \cdot \left|im\right| - 2\right)\right) \cdot \left(1 + \left|im\right| \cdot \left(1 + \frac{1}{2} \cdot \left|im\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (* (* re re) re) re)))
  (if (<=
       (* (* 1/2 (cos re)) (+ (exp (- (fabs im))) (exp (fabs im))))
       -3602879701896397/72057594037927936)
    (* (+ 1/2 (* -1/4 (sqrt (sqrt (* t_0 t_0))))) 2)
    (*
     1/2
     (*
      (+ 2 (* (fabs im) (- (* 2 (fabs im)) 2)))
      (+ 1 (* (fabs im) (+ 1 (* 1/2 (fabs im))))))))))

C

double code(double re, double im) {
	double t_0 = ((re * re) * re) * re;
	double tmp;
	if (((0.5 * cos(re)) * (exp(-fabs(im)) + exp(fabs(im)))) <= -0.05) {
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_0 * t_0))))) * 2.0;
	} else {
		tmp = 0.5 * ((2.0 + (fabs(im) * ((2.0 * fabs(im)) - 2.0))) * (1.0 + (fabs(im) * (1.0 + (0.5 * fabs(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) :: t_0
    real(8) :: tmp
    t_0 = ((re * re) * re) * re
    if (((0.5d0 * cos(re)) * (exp(-abs(im)) + exp(abs(im)))) <= (-0.05d0)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(sqrt((t_0 * t_0))))) * 2.0d0
    else
        tmp = 0.5d0 * ((2.0d0 + (abs(im) * ((2.0d0 * abs(im)) - 2.0d0))) * (1.0d0 + (abs(im) * (1.0d0 + (0.5d0 * abs(im))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((re * re) * re) * re;
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-Math.abs(im)) + Math.exp(Math.abs(im)))) <= -0.05) {
		tmp = (0.5 + (-0.25 * Math.sqrt(Math.sqrt((t_0 * t_0))))) * 2.0;
	} else {
		tmp = 0.5 * ((2.0 + (Math.abs(im) * ((2.0 * Math.abs(im)) - 2.0))) * (1.0 + (Math.abs(im) * (1.0 + (0.5 * Math.abs(im))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((re * re) * re) * re
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-math.fabs(im)) + math.exp(math.fabs(im)))) <= -0.05:
		tmp = (0.5 + (-0.25 * math.sqrt(math.sqrt((t_0 * t_0))))) * 2.0
	else:
		tmp = 0.5 * ((2.0 + (math.fabs(im) * ((2.0 * math.fabs(im)) - 2.0))) * (1.0 + (math.fabs(im) * (1.0 + (0.5 * math.fabs(im))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(re * re) * re) * re)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-abs(im))) + exp(abs(im)))) <= -0.05)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(sqrt(Float64(t_0 * t_0))))) * 2.0);
	else
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(abs(im) * Float64(Float64(2.0 * abs(im)) - 2.0))) * Float64(1.0 + Float64(abs(im) * Float64(1.0 + Float64(0.5 * abs(im)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((re * re) * re) * re;
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-abs(im)) + exp(abs(im)))) <= -0.05)
		tmp = (0.5 + (-0.25 * sqrt(sqrt((t_0 * t_0))))) * 2.0;
	else
		tmp = 0.5 * ((2.0 + (abs(im) * ((2.0 * abs(im)) - 2.0))) * (1.0 + (abs(im) * (1.0 + (0.5 * abs(im))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] + N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3602879701896397/72057594037927936], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(1/2 * N[(N[(2 + N[(N[Abs[im], $MachinePrecision] * N[(N[(2 * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1 + N[(N[Abs[im], $MachinePrecision] * N[(1 + N[(1/2 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}\right) \cdot 2 \]

       2. sqrt-unprod N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       4. lower-*.f64 37.0%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       7. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       11. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)}}\right) \cdot 2 \]

       14. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)}}\right) \cdot 2 \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot re\right)}}\right) \cdot 2 \]

       17. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

       18. lower-*.f64 37.0%

           \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   11. Applied rewrites 37.0%

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\sqrt{\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot re\right)}}\right) \cdot 2 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 55.8%

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

Alternative 9: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-\left|im\right|} + e^{\left|im\right|}\right) \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(2 + \left|im\right| \cdot \left(2 \cdot \left|im\right| - 2\right)\right) \cdot \left(1 + \left|im\right| \cdot \left(1 + \frac{1}{2} \cdot \left|im\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     (* (* 1/2 (cos re)) (+ (exp (- (fabs im))) (exp (fabs im))))
     -3602879701896397/72057594037927936)
  (* (+ 1/2 (* -1/4 (sqrt (* (* re re) (* re re))))) 2)
  (*
   1/2
   (*
    (+ 2 (* (fabs im) (- (* 2 (fabs im)) 2)))
    (+ 1 (* (fabs im) (+ 1 (* 1/2 (fabs im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-fabs(im)) + exp(fabs(im)))) <= -0.05) {
		tmp = (0.5 + (-0.25 * sqrt(((re * re) * (re * re))))) * 2.0;
	} else {
		tmp = 0.5 * ((2.0 + (fabs(im) * ((2.0 * fabs(im)) - 2.0))) * (1.0 + (fabs(im) * (1.0 + (0.5 * fabs(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 * cos(re)) * (exp(-abs(im)) + exp(abs(im)))) <= (-0.05d0)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(((re * re) * (re * re))))) * 2.0d0
    else
        tmp = 0.5d0 * ((2.0d0 + (abs(im) * ((2.0d0 * abs(im)) - 2.0d0))) * (1.0d0 + (abs(im) * (1.0d0 + (0.5d0 * abs(im))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-Math.abs(im)) + Math.exp(Math.abs(im)))) <= -0.05) {
		tmp = (0.5 + (-0.25 * Math.sqrt(((re * re) * (re * re))))) * 2.0;
	} else {
		tmp = 0.5 * ((2.0 + (Math.abs(im) * ((2.0 * Math.abs(im)) - 2.0))) * (1.0 + (Math.abs(im) * (1.0 + (0.5 * Math.abs(im))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-math.fabs(im)) + math.exp(math.fabs(im)))) <= -0.05:
		tmp = (0.5 + (-0.25 * math.sqrt(((re * re) * (re * re))))) * 2.0
	else:
		tmp = 0.5 * ((2.0 + (math.fabs(im) * ((2.0 * math.fabs(im)) - 2.0))) * (1.0 + (math.fabs(im) * (1.0 + (0.5 * math.fabs(im))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-abs(im))) + exp(abs(im)))) <= -0.05)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(Float64(Float64(re * re) * Float64(re * re))))) * 2.0);
	else
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(abs(im) * Float64(Float64(2.0 * abs(im)) - 2.0))) * Float64(1.0 + Float64(abs(im) * Float64(1.0 + Float64(0.5 * abs(im)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-abs(im)) + exp(abs(im)))) <= -0.05)
		tmp = (0.5 + (-0.25 * sqrt(((re * re) * (re * re))))) * 2.0;
	else
		tmp = 0.5 * ((2.0 + (abs(im) * ((2.0 * abs(im)) - 2.0))) * (1.0 + (abs(im) * (1.0 + (0.5 * abs(im))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] + N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3602879701896397/72057594037927936], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(1/2 * N[(N[(2 + N[(N[Abs[im], $MachinePrecision] * N[(N[(2 * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1 + N[(N[Abs[im], $MachinePrecision] * N[(1 + N[(1/2 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 87.6%

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

   9. Applied rewrites 87.6%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 55.8%

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

Alternative 10: 49.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2\\ \mathbf{elif}\;t\_0 \leq 10000000000000000373409337471459889719393275754491820381027730410378005080671497101378613371421126415052399029342192009216:\\ \;\;\;\;\frac{1}{2} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \frac{2}{re \cdot re}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im)))))
  (if (<= t_0 -3602879701896397/72057594037927936)
    (* (+ 1/2 (* -1/4 (sqrt (* (* re re) (* re re))))) 2)
    (if (<=
         t_0
         10000000000000000373409337471459889719393275754491820381027730410378005080671497101378613371421126415052399029342192009216)
      (* 1/2 2)
      (* (* (* (- 1 (/ 2 (* re re))) (* -1/4 re)) re) 2)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (0.5 + (-0.25 * sqrt(((re * re) * (re * re))))) * 2.0;
	} else if (t_0 <= 1e+121) {
		tmp = 0.5 * 2.0;
	} else {
		tmp = (((1.0 - (2.0 / (re * re))) * (-0.25 * re)) * 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 * cos(re)) * (exp(-im) + exp(im))
    if (t_0 <= (-0.05d0)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(((re * re) * (re * re))))) * 2.0d0
    else if (t_0 <= 1d+121) then
        tmp = 0.5d0 * 2.0d0
    else
        tmp = (((1.0d0 - (2.0d0 / (re * re))) * ((-0.25d0) * re)) * re) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (0.5 + (-0.25 * Math.sqrt(((re * re) * (re * re))))) * 2.0;
	} else if (t_0 <= 1e+121) {
		tmp = 0.5 * 2.0;
	} else {
		tmp = (((1.0 - (2.0 / (re * re))) * (-0.25 * re)) * re) * 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -0.05:
		tmp = (0.5 + (-0.25 * math.sqrt(((re * re) * (re * re))))) * 2.0
	elif t_0 <= 1e+121:
		tmp = 0.5 * 2.0
	else:
		tmp = (((1.0 - (2.0 / (re * re))) * (-0.25 * re)) * re) * 2.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(Float64(Float64(re * re) * Float64(re * re))))) * 2.0);
	elseif (t_0 <= 1e+121)
		tmp = Float64(0.5 * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(2.0 / Float64(re * re))) * Float64(-0.25 * re)) * re) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = (0.5 + (-0.25 * sqrt(((re * re) * (re * re))))) * 2.0;
	elseif (t_0 <= 1e+121)
		tmp = 0.5 * 2.0;
	else
		tmp = (((1.0 - (2.0 / (re * re))) * (-0.25 * re)) * re) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3602879701896397/72057594037927936], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[t$95$0, 10000000000000000373409337471459889719393275754491820381027730410378005080671497101378613371421126415052399029342192009216], N[(1/2 * 2), $MachinePrecision], N[(N[(N[(N[(1 - N[(2 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1/4 * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * 2), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1e121

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]

   if 1e121 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \cdot 2 \]

      3. add-flip N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \cdot 2 \]

      4. sub-to-mult N/A

         \[\leadsto \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(\color{blue}{\frac{-1}{4}} \cdot {re}^{2}\right)\right) \cdot 2 \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right) \cdot 2 \]

      8. metadata-eval 21.1%

         \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right) \cdot 2 \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right) \cdot 2 \]

      10. unpow2 N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right) \cdot 2 \]

      11. lower-*.f64 21.1%

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right) \cdot 2 \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

      13. unpow2 N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

      14. lower-*.f64 21.1%

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

   9. Applied rewrites 21.1%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(re \cdot re\right)\right)}\right) \cdot 2 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\left(\frac{-1}{4} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot \color{blue}{re}\right) \cdot 2 \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot \color{blue}{re}\right) \cdot 2 \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

       8. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(1 - \frac{\frac{-1}{2}}{\frac{-1}{4} \cdot \left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

       10. associate-/r* N/A

           \[\leadsto \left(\left(\left(1 - \frac{\frac{\frac{-1}{2}}{\frac{-1}{4}}}{re \cdot re}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(\left(1 - \frac{2}{re \cdot re}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

       12. lower-/.f64 N/A

           \[\leadsto \left(\left(\left(1 - \frac{2}{re \cdot re}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

       13. lower-*.f64 32.9%

           \[\leadsto \left(\left(\left(1 - \frac{2}{re \cdot re}\right) \cdot \left(\frac{-1}{4} \cdot re\right)\right) \cdot re\right) \cdot 2 \]

   11. Applied rewrites 32.9%

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

Alternative 11: 38.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot 2\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im)))
     -3602879701896397/72057594037927936)
  (* (+ 1/2 (* -1/4 (sqrt (* (* re re) (* re re))))) 2)
  (* 1/2 2)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = (0.5 + (-0.25 * sqrt(((re * re) * (re * re))))) * 2.0;
	} else {
		tmp = 0.5 * 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 * cos(re)) * (exp(-im) + exp(im))) <= (-0.05d0)) then
        tmp = (0.5d0 + ((-0.25d0) * sqrt(((re * re) * (re * re))))) * 2.0d0
    else
        tmp = 0.5d0 * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im))) <= -0.05) {
		tmp = (0.5 + (-0.25 * Math.sqrt(((re * re) * (re * re))))) * 2.0;
	} else {
		tmp = 0.5 * 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= -0.05:
		tmp = (0.5 + (-0.25 * math.sqrt(((re * re) * (re * re))))) * 2.0
	else:
		tmp = 0.5 * 2.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * sqrt(Float64(Float64(re * re) * Float64(re * re))))) * 2.0);
	else
		tmp = Float64(0.5 * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05)
		tmp = (0.5 + (-0.25 * sqrt(((re * re) * (re * re))))) * 2.0;
	else
		tmp = 0.5 * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3602879701896397/72057594037927936], N[(N[(1/2 + N[(-1/4 * N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(1/2 * 2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \left(\sqrt{{re}^{2}} \cdot \color{blue}{\sqrt{{re}^{2}}}\right)\right) \cdot 2 \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      4. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{{re}^{2} \cdot {re}^{2}}\right) \cdot 2 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      7. lower-*.f64 35.5%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot {re}^{2}}\right) \cdot 2 \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

      10. lower-*.f64 35.5%

          \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   9. Applied rewrites 35.5%

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right) \cdot 2 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;2 \cdot \left(\frac{-1}{4} \cdot \left(re \cdot re\right) - \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot 2\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im)))
     -3602879701896397/72057594037927936)
  (* 2 (- (* -1/4 (* re re)) -1/2))
  (* 1/2 2)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = 2.0 * ((-0.25 * (re * re)) - -0.5);
	} else {
		tmp = 0.5 * 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 * cos(re)) * (exp(-im) + exp(im))) <= (-0.05d0)) then
        tmp = 2.0d0 * (((-0.25d0) * (re * re)) - (-0.5d0))
    else
        tmp = 0.5d0 * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im))) <= -0.05) {
		tmp = 2.0 * ((-0.25 * (re * re)) - -0.5);
	} else {
		tmp = 0.5 * 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= -0.05:
		tmp = 2.0 * ((-0.25 * (re * re)) - -0.5)
	else:
		tmp = 0.5 * 2.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(2.0 * Float64(Float64(-0.25 * Float64(re * re)) - -0.5));
	else
		tmp = Float64(0.5 * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05)
		tmp = 2.0 * ((-0.25 * (re * re)) - -0.5);
	else
		tmp = 0.5 * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(1/2 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3602879701896397/72057594037927936], N[(2 * N[(N[(-1/4 * N[(re * re), $MachinePrecision]), $MachinePrecision] - -1/2), $MachinePrecision]), $MachinePrecision], N[(1/2 * 2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.0%

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 33.0%

         \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \]

      6. add-flip N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      7. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot \left(re \cdot re\right) - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot \left(re \cdot re\right) - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      11. metadata-eval 33.0%

          \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot \left(re \cdot re\right) - \frac{-1}{2}\right) \]

   9. Applied rewrites 33.0%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{4} \cdot \left(re \cdot re\right) - \frac{-1}{2}\right)} \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.2%

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 28.6% accurate, 52.7× speedup

Math

\[\frac{1}{2} \cdot 2 \]

FPCore

(FPCore (re im)
  :precision binary64
  (* 1/2 2))

C

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

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 * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1/2 * 2), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.2%

   \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
6. Step-by-step derivation

7. Applied rewrites 28.6%

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 1/2 (cos re)) (+ (exp (- im)) (exp im))))