math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.4×

• 49.5% 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: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \mathsf{fma}\left(\frac{\sin re}{e^{im\_m}}, 0.5, e^{im\_m} \cdot \left(\sin re \cdot 0.5\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (fma (/ (sin re) (exp im_m)) 0.5 (* (exp im_m) (* (sin re) 0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return fma((sin(re) / exp(im_m)), 0.5, (exp(im_m) * (sin(re) * 0.5)));
}

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(Float64(sin(re) / exp(im_m)), 0.5, Float64(exp(im_m) * Float64(sin(re) * 0.5)))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(N[Sin[re], $MachinePrecision] / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Exp[im$95$m], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot e^{0 - im} + \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im}} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot e^{0 - im} + \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im} \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot e^{0 - im}\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot e^{0 - im}\right) \cdot \frac{1}{2}} + \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot e^{0 - im}, \frac{1}{2}, \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im}\right)} \]

   8. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re \cdot \color{blue}{e^{0 - im}}, \frac{1}{2}, \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im}\right) \]

   9. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re \cdot e^{\color{blue}{0 - im}}, \frac{1}{2}, \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im}\right) \]

   10. sub0-neg N/A

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

   11. exp-neg N/A

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

   12. lift-exp.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left(\frac{\sin re}{e^{im}}, 0.5, \color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left(\frac{\sin re}{e^{im}}, 0.5, e^{im} \cdot \color{blue}{\left(\sin re \cdot 0.5\right)}\right) \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin re}{e^{im}}, 0.5, e^{im} \cdot \left(\sin re \cdot 0.5\right)\right)} \]
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sin re \cdot \cosh im\_m \end{array} \]

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sin(re) * cosh(im_m);
}

Fortran

im_m =     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, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sin(re) * cosh(im_m)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sin(re) * Math.cosh(im_m);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sin(re) * math.cosh(im_m)

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(sin(re) * cosh(im_m))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sin(re) * cosh(im_m);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

   8. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. *-lft-identity N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

3. Applied rewrites 100.0%

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

Alternative 3: 86.8% accurate, 0.4× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im_m)) (exp im_m)))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* (* re re) re) -0.16666666666666666 re) (cosh im_m))
     (if (<= t_0 1.0) (* (* (sin re) 2.0) 0.5) (* re (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im_m)) + exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * re), -0.16666666666666666, re) * cosh(im_m);
	} else if (t_0 <= 1.0) {
		tmp = (sin(re) * 2.0) * 0.5;
	} else {
		tmp = re * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * re), -0.16666666666666666, re) * cosh(im_m));
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(sin(re) * 2.0) * 0.5);
	else
		tmp = Float64(re * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[Sin[re], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(re * N[Cosh[im$95$m], $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))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9

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

   6. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \cosh im \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right) \cdot \cosh im \]

      12. lower-*.f64 62.9

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot re, -0.16666666666666666, re\right) \cdot \cosh im \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right) \cdot \cosh im \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, \frac{-1}{6}, re\right) \cdot \cosh im \]

      15. lower-*.f64 62.9

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, -0.16666666666666666, re\right) \cdot \cosh im \]

   8. Applied rewrites 62.9%

      \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, \color{blue}{-0.16666666666666666}, re\right) \cdot \cosh im \]

   if -inf.0 < (*.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

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

   4. Applied rewrites 50.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 61.9%

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

Alternative 4: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im\_m} + e^{im\_m}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot re, -0.16666666666666666, re\right) \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im_m)) (exp im_m))) 5e-7)
   (* (fma (* (* re re) re) -0.16666666666666666 re) (cosh im_m))
   (* re (cosh im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im_m)) + exp(im_m))) <= 5e-7) {
		tmp = fma(((re * re) * re), -0.16666666666666666, re) * cosh(im_m);
	} else {
		tmp = re * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im_m)) + exp(im_m))) <= 5e-7)
		tmp = Float64(fma(Float64(Float64(re * re) * re), -0.16666666666666666, re) * cosh(im_m));
	else
		tmp = Float64(re * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < 4.99999999999999977e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9

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

   6. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \cosh im \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right) \cdot \cosh im \]

      12. lower-*.f64 62.9

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot re, -0.16666666666666666, re\right) \cdot \cosh im \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right) \cdot \cosh im \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, \frac{-1}{6}, re\right) \cdot \cosh im \]

      15. lower-*.f64 62.9

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, -0.16666666666666666, re\right) \cdot \cosh im \]

   8. Applied rewrites 62.9%

      \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, \color{blue}{-0.16666666666666666}, re\right) \cdot \cosh im \]

   if 4.99999999999999977e-7 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 61.9%

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

Alternative 5: 47.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im\_m} + e^{im\_m}\right) \leq -0.96:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(1 + \frac{1 \cdot 1 - im\_m \cdot im\_m}{1 + im\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im_m)) (exp im_m))) -0.96)
   (* 0.5 (* re (+ 1.0 (/ (- (* 1.0 1.0) (* im_m im_m)) (+ 1.0 im_m)))))
   (* re (cosh im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im_m)) + exp(im_m))) <= -0.96) {
		tmp = 0.5 * (re * (1.0 + (((1.0 * 1.0) - (im_m * im_m)) / (1.0 + im_m))));
	} else {
		tmp = re * cosh(im_m);
	}
	return tmp;
}

Fortran

im_m =     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, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp((0.0d0 - im_m)) + exp(im_m))) <= (-0.96d0)) then
        tmp = 0.5d0 * (re * (1.0d0 + (((1.0d0 * 1.0d0) - (im_m * im_m)) / (1.0d0 + im_m))))
    else
        tmp = re * cosh(im_m)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp((0.0 - im_m)) + Math.exp(im_m))) <= -0.96) {
		tmp = 0.5 * (re * (1.0 + (((1.0 * 1.0) - (im_m * im_m)) / (1.0 + im_m))));
	} else {
		tmp = re * Math.cosh(im_m);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp((0.0 - im_m)) + math.exp(im_m))) <= -0.96:
		tmp = 0.5 * (re * (1.0 + (((1.0 * 1.0) - (im_m * im_m)) / (1.0 + im_m))))
	else:
		tmp = re * math.cosh(im_m)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im_m)) + exp(im_m))) <= -0.96)
		tmp = Float64(0.5 * Float64(re * Float64(1.0 + Float64(Float64(Float64(1.0 * 1.0) - Float64(im_m * im_m)) / Float64(1.0 + im_m)))));
	else
		tmp = Float64(re * cosh(im_m));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp((0.0 - im_m)) + exp(im_m))) <= -0.96)
		tmp = 0.5 * (re * (1.0 + (((1.0 * 1.0) - (im_m * im_m)) / (1.0 + im_m))));
	else
		tmp = re * cosh(im_m);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.96], N[(0.5 * N[(re * N[(1.0 + N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < -0.95999999999999996

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 31.5

         \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + -1 \cdot im\right)\right)\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{-1 \cdot im}\right)\right)\right) \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

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

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

       4. flip-- N/A

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

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

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

       6. lift-*.f64 N/A

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

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

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

       8. lift-*.f64 N/A

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

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

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

       10. lift-*.f64 N/A

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

       11. lower-special-+.f32 N/A

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

       12. lower-+.f32 N/A

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

       13. lower-special-/.f64 N/A

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

   12. Applied rewrites 32.9%

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

   if -0.95999999999999996 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 61.9%

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

Alternative 6: 47.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im\_m} + e^{im\_m}\right) \leq -0.8:\\ \;\;\;\;\left(\left(1 - im\_m\right) + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im_m)) (exp im_m))) -0.8)
   (* (+ (- 1.0 im_m) 1.0) (* 0.5 re))
   (* re (cosh im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im_m)) + exp(im_m))) <= -0.8) {
		tmp = ((1.0 - im_m) + 1.0) * (0.5 * re);
	} else {
		tmp = re * cosh(im_m);
	}
	return tmp;
}

Fortran

im_m =     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, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp((0.0d0 - im_m)) + exp(im_m))) <= (-0.8d0)) then
        tmp = ((1.0d0 - im_m) + 1.0d0) * (0.5d0 * re)
    else
        tmp = re * cosh(im_m)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp((0.0 - im_m)) + Math.exp(im_m))) <= -0.8) {
		tmp = ((1.0 - im_m) + 1.0) * (0.5 * re);
	} else {
		tmp = re * Math.cosh(im_m);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp((0.0 - im_m)) + math.exp(im_m))) <= -0.8:
		tmp = ((1.0 - im_m) + 1.0) * (0.5 * re)
	else:
		tmp = re * math.cosh(im_m)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im_m)) + exp(im_m))) <= -0.8)
		tmp = Float64(Float64(Float64(1.0 - im_m) + 1.0) * Float64(0.5 * re));
	else
		tmp = Float64(re * cosh(im_m));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp((0.0 - im_m)) + exp(im_m))) <= -0.8)
		tmp = ((1.0 - im_m) + 1.0) * (0.5 * re);
	else
		tmp = re * cosh(im_m);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < -0.80000000000000004

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 31.5

         \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + -1 \cdot im\right)\right)\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{-1 \cdot im}\right)\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. lift-+.f64 N/A

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

       10. add-flip N/A

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

       11. lower--.f64 N/A

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

       12. lift-*.f64 N/A

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

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

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

       14. metadata-eval N/A

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

       15. *-lft-identity N/A

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

       16. *-lft-identity N/A

           \[\leadsto \left(\left(1 - im\right) + 1\right) \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

   12. Applied rewrites 31.5%

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

   if -0.80000000000000004 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 61.9%

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

Alternative 7: 31.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \left(\left(1 - im\_m\right) + 1\right) \cdot \left(0.5 \cdot re\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (+ (- 1.0 im_m) 1.0) (* 0.5 re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return ((1.0 - im_m) + 1.0) * (0.5 * re);
}

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return ((1.0 - im_m) + 1.0) * (0.5 * re);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return ((1.0 - im_m) + 1.0) * (0.5 * re)

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(Float64(1.0 - im_m) + 1.0) * Float64(0.5 * re))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = ((1.0 - im_m) + 1.0) * (0.5 * re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(N[(1.0 - im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $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

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 61.9

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 25.8%

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

8. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 31.5

      \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + -1 \cdot im\right)\right)\right) \]

10. Applied rewrites 31.5%

    \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{-1 \cdot im}\right)\right)\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lift-+.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. add-flip N/A

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

    11. lower--.f64 N/A

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

    12. lift-*.f64 N/A

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

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

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

    14. metadata-eval N/A

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

    15. *-lft-identity N/A

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

    16. *-lft-identity N/A

        \[\leadsto \left(\left(1 - im\right) + 1\right) \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

12. Applied rewrites 31.5%

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

Alternative 8: 26.1% accurate, 9.3× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (* 0.5 re) 2.0))

C

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

Fortran

im_m =     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, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = (0.5d0 * re) * 2.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return (0.5 * re) * 2.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return (0.5 * re) * 2.0

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * re), $MachinePrecision] * 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 im around 0

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 26.1%

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

Reproduce

herbie shell --seed 2025151 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))