math.sin on complex, real part

Percentage Accurate: 100.0% → 75.3%

Time: 4.1s

Alternatives: 9

Speedup: N/A×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\_m\\ t_1 := t\_0 \cdot \left(e^{0 - im\_m} + e^{im\_m}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.998:\\ \;\;\;\;\left(\left(\cosh im\_m \cdot 2\right) \cdot \left(\left(-0.16666666666666666 - 2\right) \cdot -0.16666666666666666\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot \left(0.5 + -0.08333333333333333 \cdot {re\_m}^{2}\right)\right) \cdot \left(1 + e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re_m)))
        (t_1 (* t_0 (+ (exp (- 0.0 im_m)) (exp im_m)))))
   (*
    re_s
    (if (<= t_1 -0.998)
      (*
       (*
        (* (cosh im_m) 2.0)
        (* (- -0.16666666666666666 2.0) -0.16666666666666666))
       0.5)
      (if (<= t_1 1.0)
        (* t_0 2.0)
        (*
         (* re_m (+ 0.5 (* -0.08333333333333333 (pow re_m 2.0))))
         (+ 1.0 (exp im_m))))))))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	double t_0 = 0.5 * sin(re_m);
	double t_1 = t_0 * (exp((0.0 - im_m)) + exp(im_m));
	double tmp;
	if (t_1 <= -0.998) {
		tmp = ((cosh(im_m) * 2.0) * ((-0.16666666666666666 - 2.0) * -0.16666666666666666)) * 0.5;
	} else if (t_1 <= 1.0) {
		tmp = t_0 * 2.0;
	} else {
		tmp = (re_m * (0.5 + (-0.08333333333333333 * pow(re_m, 2.0)))) * (1.0 + exp(im_m));
	}
	return re_s * tmp;
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re_m)
    t_1 = t_0 * (exp((0.0d0 - im_m)) + exp(im_m))
    if (t_1 <= (-0.998d0)) then
        tmp = ((cosh(im_m) * 2.0d0) * (((-0.16666666666666666d0) - 2.0d0) * (-0.16666666666666666d0))) * 0.5d0
    else if (t_1 <= 1.0d0) then
        tmp = t_0 * 2.0d0
    else
        tmp = (re_m * (0.5d0 + ((-0.08333333333333333d0) * (re_m ** 2.0d0)))) * (1.0d0 + exp(im_m))
    end if
    code = re_s * tmp
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	double t_0 = 0.5 * Math.sin(re_m);
	double t_1 = t_0 * (Math.exp((0.0 - im_m)) + Math.exp(im_m));
	double tmp;
	if (t_1 <= -0.998) {
		tmp = ((Math.cosh(im_m) * 2.0) * ((-0.16666666666666666 - 2.0) * -0.16666666666666666)) * 0.5;
	} else if (t_1 <= 1.0) {
		tmp = t_0 * 2.0;
	} else {
		tmp = (re_m * (0.5 + (-0.08333333333333333 * Math.pow(re_m, 2.0)))) * (1.0 + Math.exp(im_m));
	}
	return re_s * tmp;
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	t_0 = 0.5 * math.sin(re_m)
	t_1 = t_0 * (math.exp((0.0 - im_m)) + math.exp(im_m))
	tmp = 0
	if t_1 <= -0.998:
		tmp = ((math.cosh(im_m) * 2.0) * ((-0.16666666666666666 - 2.0) * -0.16666666666666666)) * 0.5
	elif t_1 <= 1.0:
		tmp = t_0 * 2.0
	else:
		tmp = (re_m * (0.5 + (-0.08333333333333333 * math.pow(re_m, 2.0)))) * (1.0 + math.exp(im_m))
	return re_s * tmp

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	t_0 = Float64(0.5 * sin(re_m))
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_1 <= -0.998)
		tmp = Float64(Float64(Float64(cosh(im_m) * 2.0) * Float64(Float64(-0.16666666666666666 - 2.0) * -0.16666666666666666)) * 0.5);
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(Float64(re_m * Float64(0.5 + Float64(-0.08333333333333333 * (re_m ^ 2.0)))) * Float64(1.0 + exp(im_m)));
	end
	return Float64(re_s * tmp)
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp_2 = code(re_s, re_m, im_m)
	t_0 = 0.5 * sin(re_m);
	t_1 = t_0 * (exp((0.0 - im_m)) + exp(im_m));
	tmp = 0.0;
	if (t_1 <= -0.998)
		tmp = ((cosh(im_m) * 2.0) * ((-0.16666666666666666 - 2.0) * -0.16666666666666666)) * 0.5;
	elseif (t_1 <= 1.0)
		tmp = t_0 * 2.0;
	else
		tmp = (re_m * (0.5 + (-0.08333333333333333 * (re_m ^ 2.0)))) * (1.0 + exp(im_m));
	end
	tmp_2 = re_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(re$95$s * If[LessEqual[t$95$1, -0.998], N[(N[(N[(N[Cosh[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(-0.16666666666666666 - 2.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[(re$95$m * N[(0.5 + N[(-0.08333333333333333 * N[Power[re$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.998

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 63.7%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)}\right) \cdot 0.5 \]

   8. Applied rewrites 41.5%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \left(\left(-0.16666666666666666 - 2\right) \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot 0.5 \]

   if -0.998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.2%

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

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(1 + e^{im}\right) \]

   7. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := \cosh im\_m \cdot 2\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.05 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \sin re\_m\right) \cdot 2\\ \mathbf{elif}\;im\_m \leq 1.6 \cdot 10^{+181}:\\ \;\;\;\;\left(t\_0 \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(-0.16666666666666666 - 2\right)\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (let* ((t_0 (* (cosh im_m) 2.0)))
   (*
    re_s
    (if (<= im_m 3.05e-5)
      (* (* 0.5 (sin re_m)) 2.0)
      (if (<= im_m 1.6e+181)
        (* (* t_0 (* -0.16666666666666666 -0.16666666666666666)) 0.5)
        (* (* t_0 (- -0.16666666666666666 2.0)) 0.5))))))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	double t_0 = cosh(im_m) * 2.0;
	double tmp;
	if (im_m <= 3.05e-5) {
		tmp = (0.5 * sin(re_m)) * 2.0;
	} else if (im_m <= 1.6e+181) {
		tmp = (t_0 * (-0.16666666666666666 * -0.16666666666666666)) * 0.5;
	} else {
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5;
	}
	return re_s * tmp;
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(im_m) * 2.0d0
    if (im_m <= 3.05d-5) then
        tmp = (0.5d0 * sin(re_m)) * 2.0d0
    else if (im_m <= 1.6d+181) then
        tmp = (t_0 * ((-0.16666666666666666d0) * (-0.16666666666666666d0))) * 0.5d0
    else
        tmp = (t_0 * ((-0.16666666666666666d0) - 2.0d0)) * 0.5d0
    end if
    code = re_s * tmp
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	double t_0 = Math.cosh(im_m) * 2.0;
	double tmp;
	if (im_m <= 3.05e-5) {
		tmp = (0.5 * Math.sin(re_m)) * 2.0;
	} else if (im_m <= 1.6e+181) {
		tmp = (t_0 * (-0.16666666666666666 * -0.16666666666666666)) * 0.5;
	} else {
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5;
	}
	return re_s * tmp;
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	t_0 = math.cosh(im_m) * 2.0
	tmp = 0
	if im_m <= 3.05e-5:
		tmp = (0.5 * math.sin(re_m)) * 2.0
	elif im_m <= 1.6e+181:
		tmp = (t_0 * (-0.16666666666666666 * -0.16666666666666666)) * 0.5
	else:
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5
	return re_s * tmp

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	t_0 = Float64(cosh(im_m) * 2.0)
	tmp = 0.0
	if (im_m <= 3.05e-5)
		tmp = Float64(Float64(0.5 * sin(re_m)) * 2.0);
	elseif (im_m <= 1.6e+181)
		tmp = Float64(Float64(t_0 * Float64(-0.16666666666666666 * -0.16666666666666666)) * 0.5);
	else
		tmp = Float64(Float64(t_0 * Float64(-0.16666666666666666 - 2.0)) * 0.5);
	end
	return Float64(re_s * tmp)
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp_2 = code(re_s, re_m, im_m)
	t_0 = cosh(im_m) * 2.0;
	tmp = 0.0;
	if (im_m <= 3.05e-5)
		tmp = (0.5 * sin(re_m)) * 2.0;
	elseif (im_m <= 1.6e+181)
		tmp = (t_0 * (-0.16666666666666666 * -0.16666666666666666)) * 0.5;
	else
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5;
	end
	tmp_2 = re_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := Block[{t$95$0 = N[(N[Cosh[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, N[(re$95$s * If[LessEqual[im$95$m, 3.05e-5], N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[im$95$m, 1.6e+181], N[(N[(t$95$0 * N[(-0.16666666666666666 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 * N[(-0.16666666666666666 - 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 3.04999999999999994e-5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.2%

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

   if 3.04999999999999994e-5 < im < 1.6e181

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 63.7%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)}\right) \cdot 0.5 \]

   8. Applied rewrites 41.1%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \left(-0.16666666666666666 \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot 0.5 \]

   if 1.6e181 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 63.7%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)}\right) \cdot 0.5 \]

   8. Applied rewrites 15.5%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \color{blue}{\left(-0.16666666666666666 - 2\right)}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := \cosh im\_m \cdot 2\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re\_m \leq 0.02:\\ \;\;\;\;\left(t\_0 \cdot re\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(-0.16666666666666666 - 2\right)\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (let* ((t_0 (* (cosh im_m) 2.0)))
   (*
    re_s
    (if (<= (* 0.5 (sin re_m)) 0.02)
      (* (* t_0 re_m) 0.5)
      (* (* t_0 (- -0.16666666666666666 2.0)) 0.5)))))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	double t_0 = cosh(im_m) * 2.0;
	double tmp;
	if ((0.5 * sin(re_m)) <= 0.02) {
		tmp = (t_0 * re_m) * 0.5;
	} else {
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5;
	}
	return re_s * tmp;
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(im_m) * 2.0d0
    if ((0.5d0 * sin(re_m)) <= 0.02d0) then
        tmp = (t_0 * re_m) * 0.5d0
    else
        tmp = (t_0 * ((-0.16666666666666666d0) - 2.0d0)) * 0.5d0
    end if
    code = re_s * tmp
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	double t_0 = Math.cosh(im_m) * 2.0;
	double tmp;
	if ((0.5 * Math.sin(re_m)) <= 0.02) {
		tmp = (t_0 * re_m) * 0.5;
	} else {
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5;
	}
	return re_s * tmp;
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	t_0 = math.cosh(im_m) * 2.0
	tmp = 0
	if (0.5 * math.sin(re_m)) <= 0.02:
		tmp = (t_0 * re_m) * 0.5
	else:
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5
	return re_s * tmp

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	t_0 = Float64(cosh(im_m) * 2.0)
	tmp = 0.0
	if (Float64(0.5 * sin(re_m)) <= 0.02)
		tmp = Float64(Float64(t_0 * re_m) * 0.5);
	else
		tmp = Float64(Float64(t_0 * Float64(-0.16666666666666666 - 2.0)) * 0.5);
	end
	return Float64(re_s * tmp)
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp_2 = code(re_s, re_m, im_m)
	t_0 = cosh(im_m) * 2.0;
	tmp = 0.0;
	if ((0.5 * sin(re_m)) <= 0.02)
		tmp = (t_0 * re_m) * 0.5;
	else
		tmp = (t_0 * (-0.16666666666666666 - 2.0)) * 0.5;
	end
	tmp_2 = re_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := Block[{t$95$0 = N[(N[Cosh[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, N[(re$95$s * If[LessEqual[N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(t$95$0 * re$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 * N[(-0.16666666666666666 - 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 63.5%

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

   if 0.0200000000000000004 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 63.7%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)}\right) \cdot 0.5 \]

   8. Applied rewrites 15.5%

      \[\leadsto \left(\left(\cosh im \cdot 2\right) \cdot \color{blue}{\left(-0.16666666666666666 - 2\right)}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \left(\left(\left(\cosh im\_m \cdot 2\right) \cdot re\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (* re_s (* (* (* (cosh im_m) 2.0) re_m) 0.5)))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	return re_s * (((cosh(im_m) * 2.0) * re_m) * 0.5);
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = re_s * (((cosh(im_m) * 2.0d0) * re_m) * 0.5d0)
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	return re_s * (((Math.cosh(im_m) * 2.0) * re_m) * 0.5);
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	return re_s * (((math.cosh(im_m) * 2.0) * re_m) * 0.5)

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	return Float64(re_s * Float64(Float64(Float64(cosh(im_m) * 2.0) * re_m) * 0.5))
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp = code(re_s, re_m, im_m)
	tmp = re_s * (((cosh(im_m) * 2.0) * re_m) * 0.5);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * N[(N[(N[(N[Cosh[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * re$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

3. Applied rewrites 100.0%

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

4. Taylor expanded in re around 0

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

6. Applied rewrites 63.5%

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

Alternative 5: 51.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \left(\left(0.5 \cdot re\_m\right) \cdot \left(1 + e^{im\_m}\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (* re_s (* (* 0.5 re_m) (+ 1.0 (exp im_m)))))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	return re_s * ((0.5 * re_m) * (1.0 + exp(im_m)));
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = re_s * ((0.5d0 * re_m) * (1.0d0 + exp(im_m)))
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	return re_s * ((0.5 * re_m) * (1.0 + Math.exp(im_m)));
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	return re_s * ((0.5 * re_m) * (1.0 + math.exp(im_m)))

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	return Float64(re_s * Float64(Float64(0.5 * re_m) * Float64(1.0 + exp(im_m))))
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp = code(re_s, re_m, im_m)
	tmp = re_s * ((0.5 * re_m) * (1.0 + exp(im_m)));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * N[(N[(0.5 * re$95$m), $MachinePrecision] * N[(1.0 + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 62.9%

   \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
8. Add Preprocessing

Alternative 6: 32.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;re\_m \leq 1.2 \cdot 10^{+40}:\\ \;\;\;\;\left(0.5 \cdot re\_m\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot re\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (* re_s (if (<= re_m 1.2e+40) (* (* 0.5 re_m) 2.0) (* (* im_m re_m) 0.5))))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	double tmp;
	if (re_m <= 1.2e+40) {
		tmp = (0.5 * re_m) * 2.0;
	} else {
		tmp = (im_m * re_m) * 0.5;
	}
	return re_s * tmp;
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re_m <= 1.2d+40) then
        tmp = (0.5d0 * re_m) * 2.0d0
    else
        tmp = (im_m * re_m) * 0.5d0
    end if
    code = re_s * tmp
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	double tmp;
	if (re_m <= 1.2e+40) {
		tmp = (0.5 * re_m) * 2.0;
	} else {
		tmp = (im_m * re_m) * 0.5;
	}
	return re_s * tmp;
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	tmp = 0
	if re_m <= 1.2e+40:
		tmp = (0.5 * re_m) * 2.0
	else:
		tmp = (im_m * re_m) * 0.5
	return re_s * tmp

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	tmp = 0.0
	if (re_m <= 1.2e+40)
		tmp = Float64(Float64(0.5 * re_m) * 2.0);
	else
		tmp = Float64(Float64(im_m * re_m) * 0.5);
	end
	return Float64(re_s * tmp)
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp_2 = code(re_s, re_m, im_m)
	tmp = 0.0;
	if (re_m <= 1.2e+40)
		tmp = (0.5 * re_m) * 2.0;
	else
		tmp = (im_m * re_m) * 0.5;
	end
	tmp_2 = re_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * If[LessEqual[re$95$m, 1.2e+40], N[(N[(0.5 * re$95$m), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(im$95$m * re$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.2e40

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 26.7%

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

   if 1.2e40 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 63.5%

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

   8. Applied rewrites 10.3%

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

Alternative 7: 27.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;re\_m \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;\left(0.5 \cdot re\_m\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m)
 :precision binary64
 (* re_s (if (<= re_m 1.12e-18) (* (* 0.5 re_m) 2.0) (* 2.0 0.5))))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	double tmp;
	if (re_m <= 1.12e-18) {
		tmp = (0.5 * re_m) * 2.0;
	} else {
		tmp = 2.0 * 0.5;
	}
	return re_s * tmp;
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re_m <= 1.12d-18) then
        tmp = (0.5d0 * re_m) * 2.0d0
    else
        tmp = 2.0d0 * 0.5d0
    end if
    code = re_s * tmp
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	double tmp;
	if (re_m <= 1.12e-18) {
		tmp = (0.5 * re_m) * 2.0;
	} else {
		tmp = 2.0 * 0.5;
	}
	return re_s * tmp;
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	tmp = 0
	if re_m <= 1.12e-18:
		tmp = (0.5 * re_m) * 2.0
	else:
		tmp = 2.0 * 0.5
	return re_s * tmp

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	tmp = 0.0
	if (re_m <= 1.12e-18)
		tmp = Float64(Float64(0.5 * re_m) * 2.0);
	else
		tmp = Float64(2.0 * 0.5);
	end
	return Float64(re_s * tmp)
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp_2 = code(re_s, re_m, im_m)
	tmp = 0.0;
	if (re_m <= 1.12e-18)
		tmp = (0.5 * re_m) * 2.0;
	else
		tmp = 2.0 * 0.5;
	end
	tmp_2 = re_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * If[LessEqual[re$95$m, 1.12e-18], N[(N[(0.5 * re$95$m), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.12000000000000001e-18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 26.7%

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

   if 1.12000000000000001e-18 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 63.5%

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

   6. Applied rewrites 5.3%

      \[\leadsto 2 \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 5.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \left(2 \cdot 0.5\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m) :precision binary64 (* re_s (* 2.0 0.5)))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	return re_s * (2.0 * 0.5);
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = re_s * (2.0d0 * 0.5d0)
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	return re_s * (2.0 * 0.5);
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	return re_s * (2.0 * 0.5)

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	return Float64(re_s * Float64(2.0 * 0.5))
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp = code(re_s, re_m, im_m)
	tmp = re_s * (2.0 * 0.5);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * N[(2.0 * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0

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

4. Applied rewrites 63.5%

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

6. Applied rewrites 5.3%

   \[\leadsto 2 \cdot \color{blue}{0.5} \]
7. Add Preprocessing

Alternative 9: 4.9% accurate, 64.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot 2 \end{array} \]

FPCore

im_m = (fabs.f64 im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im_m) :precision binary64 (* re_s 2.0))

C

im_m = fabs(im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im_m) {
	return re_s * 2.0;
}

Fortran

im_m =     private
re\_m =     private
re\_s =     private
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_s, re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = re_s * 2.0d0
end function

Java

im_m = Math.abs(im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im_m) {
	return re_s * 2.0;
}

Python

im_m = math.fabs(im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im_m):
	return re_s * 2.0

Julia

im_m = abs(im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im_m)
	return Float64(re_s * 2.0)
end

MATLAB

im_m = abs(im);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp = code(re_s, re_m, im_m)
	tmp = re_s * 2.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0

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

4. Applied rewrites 63.5%

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

6. Applied rewrites 5.3%

   \[\leadsto 2 \cdot \color{blue}{0.5} \]

8. Applied rewrites 4.9%

   \[\leadsto \color{blue}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))