math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.8%

Time: 9.5s

Alternatives: 9

Speedup: 2.8×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 0.0004\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (or (<= t_0 -10.0) (not (<= t_0 0.0004)))
     (* t_0 t_1)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (* -0.016666666666666666 (pow im 5.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 0.0004)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + (-0.016666666666666666 * pow(im, 5.0))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = 0.5d0 * sin(re)
    if ((t_0 <= (-10.0d0)) .or. (.not. (t_0 <= 0.0004d0))) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + ((-0.016666666666666666d0) * (im ** 5.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 0.0004)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + (-0.016666666666666666 * Math.pow(im, 5.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 <= -10.0) or not (t_0 <= 0.0004):
		tmp = t_0 * t_1
	else:
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * math.pow(im, 3.0)) + (-0.016666666666666666 * math.pow(im, 5.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if ((t_0 <= -10.0) || !(t_0 <= 0.0004))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(-0.016666666666666666 * (im ^ 5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 <= -10.0) || ~((t_0 <= 0.0004)))
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + (-0.016666666666666666 * (im ^ 5.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10.0], N[Not[LessEqual[t$95$0, 0.0004]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -10 or 4.00000000000000019e-4 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -10 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4.00000000000000019e-4

   1. Initial program 28.9%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10 \lor \neg \left(e^{-im} - e^{im} \leq 0.0004\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.005 \lor \neg \left(t_0 \leq 0.0004\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.005) (not (<= t_0 0.0004)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 0.0004)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.005d0)) .or. (.not. (t_0 <= 0.0004d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 0.0004)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.005) or not (t_0 <= 0.0004):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.005) || !(t_0 <= 0.0004))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.005) || ~((t_0 <= 0.0004)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.005], N[Not[LessEqual[t$95$0, 0.0004]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.0050000000000000001 or 4.00000000000000019e-4 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

   if -0.0050000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4.00000000000000019e-4

   1. Initial program 28.4%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 99.8%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + \color{blue}{\left(-im \cdot \sin re\right)} \]

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) - im \cdot \sin re} \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re} - im \cdot \sin re \]

      5. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.005 \lor \neg \left(e^{-im} - e^{im} \leq 0.0004\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.28:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.6:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* (sin re) (* (pow im 5.0) -0.008333333333333333))))
   (if (<= im -1.3e+61)
     t_1
     (if (<= im -0.28)
       t_0
       (if (<= im 2.6)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 4.5e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	double t_1 = sin(re) * (pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -1.3e+61) {
		tmp = t_1;
	} else if (im <= -0.28) {
		tmp = t_0;
	} else if (im <= 2.6) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(-im) - exp(im)) * (0.5d0 * re)
    t_1 = sin(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    if (im <= (-1.3d+61)) then
        tmp = t_1
    else if (im <= (-0.28d0)) then
        tmp = t_0
    else if (im <= 2.6d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 4.5d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	double t_1 = Math.sin(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -1.3e+61) {
		tmp = t_1;
	} else if (im <= -0.28) {
		tmp = t_0;
	} else if (im <= 2.6) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	t_1 = math.sin(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	tmp = 0
	if im <= -1.3e+61:
		tmp = t_1
	elif im <= -0.28:
		tmp = t_0
	elif im <= 2.6:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 4.5e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re))
	t_1 = Float64(sin(re) * Float64((im ^ 5.0) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -1.3e+61)
		tmp = t_1;
	elseif (im <= -0.28)
		tmp = t_0;
	elseif (im <= 2.6)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	t_1 = sin(re) * ((im ^ 5.0) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -1.3e+61)
		tmp = t_1;
	elseif (im <= -0.28)
		tmp = t_0;
	elseif (im <= 2.6)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+61], t$95$1, If[LessEqual[im, -0.28], t$95$0, If[LessEqual[im, 2.6], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.29999999999999986e61 or 4.5e61 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 99.2%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 99.2%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      4. *-commutative 99.2%

         \[\leadsto \sin re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   if -1.29999999999999986e61 < im < -0.28000000000000003 or 2.60000000000000009 < im < 4.5e61

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 84.8%

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

      1. associate-*r* 84.8%

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

   4. Simplified 84.8%

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

   if -0.28000000000000003 < im < 2.60000000000000009

   1. Initial program 29.5%

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

   2. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 98.9%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + \color{blue}{\left(-im \cdot \sin re\right)} \]

      3. unsub-neg 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) - im \cdot \sin re} \]

      4. associate-*r* 98.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re} - im \cdot \sin re \]

      5. distribute-rgt-out-- 98.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 98.9%

         \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+61}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -0.28:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 2.6:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 4: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-im\right) \cdot {\sin re}^{-3}\\ t_1 := \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -430:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- im) (pow (sin re) -3.0)))
        (t_1 (* (sin re) (* (pow im 5.0) -0.008333333333333333))))
   (if (<= im -1.3e+61)
     t_1
     (if (<= im -430.0)
       t_0
       (if (<= im 2.5) (* (- im) (sin re)) (if (<= im 7.2e+55) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = -im * pow(sin(re), -3.0);
	double t_1 = sin(re) * (pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -1.3e+61) {
		tmp = t_1;
	} else if (im <= -430.0) {
		tmp = t_0;
	} else if (im <= 2.5) {
		tmp = -im * sin(re);
	} else if (im <= 7.2e+55) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -im * (sin(re) ** (-3.0d0))
    t_1 = sin(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    if (im <= (-1.3d+61)) then
        tmp = t_1
    else if (im <= (-430.0d0)) then
        tmp = t_0
    else if (im <= 2.5d0) then
        tmp = -im * sin(re)
    else if (im <= 7.2d+55) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -im * Math.pow(Math.sin(re), -3.0);
	double t_1 = Math.sin(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -1.3e+61) {
		tmp = t_1;
	} else if (im <= -430.0) {
		tmp = t_0;
	} else if (im <= 2.5) {
		tmp = -im * Math.sin(re);
	} else if (im <= 7.2e+55) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -im * math.pow(math.sin(re), -3.0)
	t_1 = math.sin(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	tmp = 0
	if im <= -1.3e+61:
		tmp = t_1
	elif im <= -430.0:
		tmp = t_0
	elif im <= 2.5:
		tmp = -im * math.sin(re)
	elif im <= 7.2e+55:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(-im) * (sin(re) ^ -3.0))
	t_1 = Float64(sin(re) * Float64((im ^ 5.0) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -1.3e+61)
		tmp = t_1;
	elseif (im <= -430.0)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = Float64(Float64(-im) * sin(re));
	elseif (im <= 7.2e+55)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -im * (sin(re) ^ -3.0);
	t_1 = sin(re) * ((im ^ 5.0) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -1.3e+61)
		tmp = t_1;
	elseif (im <= -430.0)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = -im * sin(re);
	elseif (im <= 7.2e+55)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-im) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+61], t$95$1, If[LessEqual[im, -430.0], t$95$0, If[LessEqual[im, 2.5], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+55], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.29999999999999986e61 or 7.19999999999999975e55 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 98.4%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 98.4%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      4. *-commutative 98.4%

         \[\leadsto \sin re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   if -1.29999999999999986e61 < im < -430 or 2.5 < im < 7.19999999999999975e55

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.3%

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

      1. associate-*r* 3.3%

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

      2. neg-mul-1 3.3%

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

   4. Simplified 3.3%

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

   6. Applied egg-rr 36.7%

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

   if -430 < im < 2.5

   1. Initial program 28.9%

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

   2. Taylor expanded in im around 0 98.3%

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

      1. associate-*r* 98.3%

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

      2. neg-mul-1 98.3%

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

   4. Simplified 98.3%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \sin re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+61}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -430:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+55}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 5: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-im\right) \cdot {\sin re}^{-3}\\ t_1 := \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -460:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 440:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- im) (pow (sin re) -3.0)))
        (t_1 (* (sin re) (* (pow im 5.0) -0.008333333333333333))))
   (if (<= im -1.3e+61)
     t_1
     (if (<= im -460.0)
       t_0
       (if (<= im 440.0)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.15e+56) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = -im * pow(sin(re), -3.0);
	double t_1 = sin(re) * (pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -1.3e+61) {
		tmp = t_1;
	} else if (im <= -460.0) {
		tmp = t_0;
	} else if (im <= 440.0) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.15e+56) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -im * (sin(re) ** (-3.0d0))
    t_1 = sin(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    if (im <= (-1.3d+61)) then
        tmp = t_1
    else if (im <= (-460.0d0)) then
        tmp = t_0
    else if (im <= 440.0d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.15d+56) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -im * Math.pow(Math.sin(re), -3.0);
	double t_1 = Math.sin(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -1.3e+61) {
		tmp = t_1;
	} else if (im <= -460.0) {
		tmp = t_0;
	} else if (im <= 440.0) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.15e+56) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -im * math.pow(math.sin(re), -3.0)
	t_1 = math.sin(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	tmp = 0
	if im <= -1.3e+61:
		tmp = t_1
	elif im <= -460.0:
		tmp = t_0
	elif im <= 440.0:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.15e+56:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(-im) * (sin(re) ^ -3.0))
	t_1 = Float64(sin(re) * Float64((im ^ 5.0) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -1.3e+61)
		tmp = t_1;
	elseif (im <= -460.0)
		tmp = t_0;
	elseif (im <= 440.0)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.15e+56)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -im * (sin(re) ^ -3.0);
	t_1 = sin(re) * ((im ^ 5.0) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -1.3e+61)
		tmp = t_1;
	elseif (im <= -460.0)
		tmp = t_0;
	elseif (im <= 440.0)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.15e+56)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-im) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+61], t$95$1, If[LessEqual[im, -460.0], t$95$0, If[LessEqual[im, 440.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+56], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.29999999999999986e61 or 1.15000000000000007e56 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 98.4%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 98.4%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      4. *-commutative 98.4%

         \[\leadsto \sin re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   if -1.29999999999999986e61 < im < -460 or 440 < im < 1.15000000000000007e56

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

   4. Simplified 2.9%

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

   6. Applied egg-rr 37.3%

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

   if -460 < im < 440

   1. Initial program 29.5%

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

   2. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 98.9%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + \color{blue}{\left(-im \cdot \sin re\right)} \]

      3. unsub-neg 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) - im \cdot \sin re} \]

      4. associate-*r* 98.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re} - im \cdot \sin re \]

      5. distribute-rgt-out-- 98.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 98.9%

         \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+61}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -460:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{elif}\;im \leq 440:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 6: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -3.15 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (* re (pow im 5.0)))))
   (if (<= im -3.15e+41)
     t_0
     (if (<= im 2.5)
       (* (- im) (sin re))
       (if (<= im 9e+55) (* (- im) (pow (sin re) -3.0)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.008333333333333333 * (re * pow(im, 5.0));
	double tmp;
	if (im <= -3.15e+41) {
		tmp = t_0;
	} else if (im <= 2.5) {
		tmp = -im * sin(re);
	} else if (im <= 9e+55) {
		tmp = -im * pow(sin(re), -3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.008333333333333333d0) * (re * (im ** 5.0d0))
    if (im <= (-3.15d+41)) then
        tmp = t_0
    else if (im <= 2.5d0) then
        tmp = -im * sin(re)
    else if (im <= 9d+55) then
        tmp = -im * (sin(re) ** (-3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.008333333333333333 * (re * Math.pow(im, 5.0));
	double tmp;
	if (im <= -3.15e+41) {
		tmp = t_0;
	} else if (im <= 2.5) {
		tmp = -im * Math.sin(re);
	} else if (im <= 9e+55) {
		tmp = -im * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.008333333333333333 * (re * math.pow(im, 5.0))
	tmp = 0
	if im <= -3.15e+41:
		tmp = t_0
	elif im <= 2.5:
		tmp = -im * math.sin(re)
	elif im <= 9e+55:
		tmp = -im * math.pow(math.sin(re), -3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.008333333333333333 * Float64(re * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -3.15e+41)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = Float64(Float64(-im) * sin(re));
	elseif (im <= 9e+55)
		tmp = Float64(Float64(-im) * (sin(re) ^ -3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.008333333333333333 * (re * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -3.15e+41)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = -im * sin(re);
	elseif (im <= 9e+55)
		tmp = -im * (sin(re) ^ -3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[(re * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.15e+41], t$95$0, If[LessEqual[im, 2.5], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9e+55], N[((-im) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.1499999999999999e41 or 8.99999999999999996e55 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 93.5%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 93.5%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 93.5%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 93.5%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      4. *-commutative 93.5%

         \[\leadsto \sin re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   5. Simplified 93.5%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Taylor expanded in re around 0 68.8%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot re\right)} \]

   if -3.1499999999999999e41 < im < 2.5

   1. Initial program 33.6%

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

   2. Taylor expanded in im around 0 92.0%

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

      1. associate-*r* 92.0%

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

      2. neg-mul-1 92.0%

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

   4. Simplified 92.0%

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

   if 2.5 < im < 8.99999999999999996e55

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.6%

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

      1. associate-*r* 3.6%

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

      2. neg-mul-1 3.6%

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

   4. Simplified 3.6%

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

   6. Applied egg-rr 35.8%

      \[\leadsto \left(-im\right) \cdot \color{blue}{{\sin re}^{-3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.15 \cdot 10^{+41}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 7: 79.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.15 \cdot 10^{+41} \lor \neg \left(im \leq 480000\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.15e+41) (not (<= im 480000.0)))
   (* -0.008333333333333333 (* re (pow im 5.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.15e+41) || !(im <= 480000.0)) {
		tmp = -0.008333333333333333 * (re * pow(im, 5.0));
	} else {
		tmp = -im * sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-3.15d+41)) .or. (.not. (im <= 480000.0d0))) then
        tmp = (-0.008333333333333333d0) * (re * (im ** 5.0d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.15e+41) || !(im <= 480000.0)) {
		tmp = -0.008333333333333333 * (re * Math.pow(im, 5.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.15e+41) or not (im <= 480000.0):
		tmp = -0.008333333333333333 * (re * math.pow(im, 5.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.15e+41) || !(im <= 480000.0))
		tmp = Float64(-0.008333333333333333 * Float64(re * (im ^ 5.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.15e+41) || ~((im <= 480000.0)))
		tmp = -0.008333333333333333 * (re * (im ^ 5.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.15e+41], N[Not[LessEqual[im, 480000.0]], $MachinePrecision]], N[(-0.008333333333333333 * N[(re * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.1499999999999999e41 or 4.8e5 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 81.4%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 81.4%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 81.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 81.4%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      4. *-commutative 81.4%

         \[\leadsto \sin re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   5. Simplified 81.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Taylor expanded in re around 0 61.4%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot re\right)} \]

   if -3.1499999999999999e41 < im < 4.8e5

   1. Initial program 34.1%

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

   2. Taylor expanded in im around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. neg-mul-1 91.4%

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

   4. Simplified 91.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.15 \cdot 10^{+41} \lor \neg \left(im \leq 480000\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 8: 56.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{+41} \lor \neg \left(im \leq 450\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.6e+41) (not (<= im 450.0)))
   (* (- im) re)
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.6e+41) || !(im <= 450.0)) {
		tmp = -im * re;
	} else {
		tmp = -im * sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-3.6d+41)) .or. (.not. (im <= 450.0d0))) then
        tmp = -im * re
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.6e+41) || !(im <= 450.0)) {
		tmp = -im * re;
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.6e+41) or not (im <= 450.0):
		tmp = -im * re
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.6e+41) || !(im <= 450.0))
		tmp = Float64(Float64(-im) * re);
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.6e+41) || ~((im <= 450.0)))
		tmp = -im * re;
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.6e+41], N[Not[LessEqual[im, 450.0]], $MachinePrecision]], N[((-im) * re), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.60000000000000025e41 or 450 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.4%

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

      1. associate-*r* 4.4%

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

      2. neg-mul-1 4.4%

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

   4. Simplified 4.4%

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

   5. Taylor expanded in re around 0 13.1%

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

   if -3.60000000000000025e41 < im < 450

   1. Initial program 34.1%

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

   2. Taylor expanded in im around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. neg-mul-1 91.4%

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

   4. Simplified 91.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{+41} \lor \neg \left(im \leq 450\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 9: 32.8% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

2. Taylor expanded in im around 0 46.2%

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

   1. associate-*r* 46.2%

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

   2. neg-mul-1 46.2%

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

4. Simplified 46.2%

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

5. Taylor expanded in re around 0 27.8%

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

6. Final simplification 27.8%

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

Developer target: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023312 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))