math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.9%

Time: 10.6s

Alternatives: 9

Speedup: 1.2×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 = ((-2.0d0 * sinh(im)) * cos(re)) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Applied rewrites 99.9%

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

Alternative 2: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2\right) \cdot \sinh im\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\left(im \cdot \cos re\right) \cdot \mathsf{fma}\left(im \cdot -0.16666666666666666, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- 2.0) (sinh im)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_1 -0.01)
     (* t_0 0.5)
     (if (<= t_1 0.0001)
       (* (* im (cos re)) (fma (* im -0.16666666666666666) im -1.0))
       (* (fma (* re -0.25) re 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = -2.0 * sinh(im);
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.0001) {
		tmp = (im * cos(re)) * fma((im * -0.16666666666666666), im, -1.0);
	} else {
		tmp = fma((re * -0.25), re, 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(-2.0) * sinh(im))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(im * cos(re)) * fma(Float64(im * -0.16666666666666666), im, -1.0));
	else
		tmp = Float64(fma(Float64(re * -0.25), re, 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * -0.16666666666666666), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0100000000000000002

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.0%

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

   6. Applied rewrites 64.9%

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

   if -0.0100000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\left(im \cdot \cos re\right) \cdot \mathsf{fma}\left(im \cdot -0.16666666666666666, im, -1\right)} \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2\right) \cdot \sinh im\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- 2.0) (sinh im)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_1 -0.01)
     (* t_0 0.5)
     (if (<= t_1 0.0001)
       (* (- im) (cos re))
       (* (fma (* re -0.25) re 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = -2.0 * sinh(im);
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.0001) {
		tmp = -im * cos(re);
	} else {
		tmp = fma((re * -0.25), re, 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(-2.0) * sinh(im))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(-im) * cos(re));
	else
		tmp = Float64(fma(Float64(re * -0.25), re, 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0100000000000000002

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.0%

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

   6. Applied rewrites 64.9%

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

   if -0.0100000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 52.1%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2\right) \cdot \sinh im\\ \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \leq 0:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- 2.0) (sinh im))))
   (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) 0.0)
     (* t_0 0.5)
     (* (fma (* re -0.25) re 0.5) t_0))))

C

double code(double re, double im) {
	double t_0 = -2.0 * sinh(im);
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 0.0) {
		tmp = t_0 * 0.5;
	} else {
		tmp = fma((re * -0.25), re, 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(-2.0) * sinh(im))
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im))) <= 0.0)
		tmp = Float64(t_0 * 0.5);
	else
		tmp = Float64(fma(Float64(re * -0.25), re, 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(t$95$0 * 0.5), $MachinePrecision], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.0%

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

   6. Applied rewrites 64.9%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) 0.0)
   (* (* (- 2.0) (sinh im)) 0.5)
   (fma -1.0 im (* 0.5 (* im (sqrt (* (* re re) (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 0.0) {
		tmp = (-2.0 * sinh(im)) * 0.5;
	} else {
		tmp = fma(-1.0, im, (0.5 * (im * sqrt(((re * re) * (re * re))))));
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(-1.0 * im + N[(0.5 * N[(im * N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.0%

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

   6. Applied rewrites 64.9%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 36.2%

      \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 51.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) 0.0)
   (* (* (- 2.0) (sinh im)) 0.5)
   (* im (fma (* 0.5 re) re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 0.0) {
		tmp = (-2.0 * sinh(im)) * 0.5;
	} else {
		tmp = im * fma((0.5 * re), re, -1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(im * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.0%

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

   6. Applied rewrites 64.9%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 35.8%

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

Alternative 7: 46.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \leq 0:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(\left(im \cdot im\right) \cdot 1\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) 0.0)
   (* im (- (* -0.16666666666666666 (* (* im im) 1.0)) 1.0))
   (* im (fma (* 0.5 re) re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 0.0) {
		tmp = im * ((-0.16666666666666666 * ((im * im) * 1.0)) - 1.0);
	} else {
		tmp = im * fma((0.5 * re), re, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im))) <= 0.0)
		tmp = Float64(im * Float64(Float64(-0.16666666666666666 * Float64(Float64(im * im) * 1.0)) - 1.0));
	else
		tmp = Float64(im * fma(Float64(0.5 * re), re, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(N[(-0.16666666666666666 * N[(N[(im * im), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 83.9%

      \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \left(\left(im \cdot im\right) \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

   7. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot 1\right) - \cos re\right) \]

   9. Applied rewrites 75.1%

      \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \left(\left(im \cdot im\right) \cdot 1\right) - \cos re\right) \]

   10. Taylor expanded in re around 0

       \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot 1\right) - 1\right) \]

   12. Applied rewrites 53.0%

       \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \left(\left(im \cdot im\right) \cdot 1\right) - 1\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 35.8%

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

Alternative 8: 33.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \leq 0:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) 0.0)
   (* -1.0 im)
   (* im (fma (* 0.5 re) re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 0.0) {
		tmp = -1.0 * im;
	} else {
		tmp = im * fma((0.5 * re), re, -1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-1.0 * im), $MachinePrecision], N[(im * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.4%

      \[\leadsto -1 \cdot im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 35.8%

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

Alternative 9: 29.4% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ -1 \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* -1.0 im))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double re, double im) {
	return -1.0 * im;
}

Python

def code(re, im):
	return -1.0 * im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 52.1%

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

5. Taylor expanded in re around 0

   \[\leadsto -1 \cdot im \]

7. Applied rewrites 29.4%

   \[\leadsto -1 \cdot im \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025153 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))