math.cos on complex, imaginary part

Percentage Accurate: 64.7% → 99.8%

Time: 9.6s

Alternatives: 13

Speedup: 2.8×

• 48.8% 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: 64.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.04\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.04)))
     (* 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 <= -1.0) || !(t_0 <= 0.04)) {
		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 <= (-1.0d0)) .or. (.not. (t_0 <= 0.04d0))) 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 <= -1.0) || !(t_0 <= 0.04)) {
		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 <= -1.0) or not (t_0 <= 0.04):
		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 <= -1.0) || !(t_0 <= 0.04))
		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 <= -1.0) || ~((t_0 <= 0.04)))
		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, -1.0], N[Not[LessEqual[t$95$0, 0.04]], $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)) < -1 or 0.0400000000000000008 < (-.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 -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0400000000000000008

   1. Initial program 37.1%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.04\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.04\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.04)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (* (pow im 5.0) -0.008333333333333333)
       (- (* (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 <= -1.0) || !(t_0 <= 0.04)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 5.0) * -0.008333333333333333) + ((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 <= (-1.0d0)) .or. (.not. (t_0 <= 0.04d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) + (((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 <= -1.0) || !(t_0 <= 0.04)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) + ((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 <= -1.0) or not (t_0 <= 0.04):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 5.0) * -0.008333333333333333) + ((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 <= -1.0) || !(t_0 <= 0.04))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) + 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 <= -1.0) || ~((t_0 <= 0.04)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 5.0) * -0.008333333333333333) + (((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, -1.0], N[Not[LessEqual[t$95$0, 0.04]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1 or 0.0400000000000000008 < (-.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 -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0400000000000000008

   1. Initial program 37.1%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

      2. neg-mul-1 99.8%

         \[\leadsto \left(\color{blue}{\left(-im \cdot \sin re\right)} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      3. distribute-lft-neg-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re}\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      5. *-commutative 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      6. distribute-rgt-in 99.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      7. +-commutative 99.8%

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 + \left(-im\right)\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      8. sub-neg 99.8%

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      9. *-commutative 99.8%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

      10. associate-*r* 99.8%

          \[\leadsto \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

      11. distribute-rgt-out 99.8%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      12. *-commutative 99.8%

          \[\leadsto \sin re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{{im}^{5} \cdot -0.008333333333333333}\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + {im}^{5} \cdot -0.008333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.04\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - im \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 2e-11)))
     (* t_0 (* 0.5 (sin re)))
     (- (* -0.16666666666666666 (* (sin re) (pow im 3.0))) (* im (sin re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 2e-11)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (-0.16666666666666666 * (sin(re) * pow(im, 3.0))) - (im * sin(re));
	}
	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 <= (-1.0d0)) .or. (.not. (t_0 <= 2d-11))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = ((-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))) - (im * sin(re))
    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 <= -1.0) || !(t_0 <= 2e-11)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (-0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0))) - (im * Math.sin(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 2e-11):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (-0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))) - (im * math.sin(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 2e-11))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0))) - Float64(im * sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 2e-11)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = (-0.16666666666666666 * (sin(re) * (im ^ 3.0))) - (im * sin(re));
	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, -1.0], N[Not[LessEqual[t$95$0, 2e-11]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1 or 1.99999999999999988e-11 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.8%

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

   if -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999988e-11

   1. Initial program 36.2%

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

   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. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - im \cdot \sin re\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 2e-11)))
     (* 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 <= -1.0) || !(t_0 <= 2e-11)) {
		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 <= (-1.0d0)) .or. (.not. (t_0 <= 2d-11))) 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 <= -1.0) || !(t_0 <= 2e-11)) {
		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 <= -1.0) or not (t_0 <= 2e-11):
		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 <= -1.0) || !(t_0 <= 2e-11))
		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 <= -1.0) || ~((t_0 <= 2e-11)))
		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, -1.0], N[Not[LessEqual[t$95$0, 2e-11]], $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)) < -1 or 1.99999999999999988e-11 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.8%

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

   if -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999988e-11

   1. Initial program 36.2%

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

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 5: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (<= t_0 -1.0)
     (* t_0 t_1)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (+
         (* -0.016666666666666666 (pow im 5.0))
         (* -0.0003968253968253968 (pow im 7.0)))))))))

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 <= -1.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.0003968253968253968 * pow(im, 7.0)))));
	}
	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 <= (-1.0d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + (((-0.016666666666666666d0) * (im ** 5.0d0)) + ((-0.0003968253968253968d0) * (im ** 7.0d0)))))
    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 <= -1.0) {
		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)) + (-0.0003968253968253968 * Math.pow(im, 7.0)))));
	}
	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 <= -1.0:
		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)) + (-0.0003968253968253968 * math.pow(im, 7.0)))))
	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 <= -1.0)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(Float64(-0.016666666666666666 * (im ^ 5.0)) + Float64(-0.0003968253968253968 * (im ^ 7.0))))));
	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 <= -1.0)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + ((-0.016666666666666666 * (im ^ 5.0)) + (-0.0003968253968253968 * (im ^ 7.0)))));
	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[LessEqual[t$95$0, -1.0], 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[(N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1

   1. Initial program 99.9%

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

   if -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 59.3%

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

   2. Taylor expanded in im around 0 98.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \]

Alternative 6: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 -1.0)
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (+
        (* (pow im 5.0) -0.008333333333333333)
        (* (pow im 7.0) -0.0001984126984126984))
       (- (* (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 <= -1.0) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * (((pow(im, 5.0) * -0.008333333333333333) + (pow(im, 7.0) * -0.0001984126984126984)) + ((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 <= (-1.0d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) + ((im ** 7.0d0) * (-0.0001984126984126984d0))) + (((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 <= -1.0) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * (((Math.pow(im, 5.0) * -0.008333333333333333) + (Math.pow(im, 7.0) * -0.0001984126984126984)) + ((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 <= -1.0:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * (((math.pow(im, 5.0) * -0.008333333333333333) + (math.pow(im, 7.0) * -0.0001984126984126984)) + ((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 <= -1.0)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(Float64((im ^ 5.0) * -0.008333333333333333) + Float64((im ^ 7.0) * -0.0001984126984126984)) + 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 <= -1.0)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((((im ^ 5.0) * -0.008333333333333333) + ((im ^ 7.0) * -0.0001984126984126984)) + (((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[LessEqual[t$95$0, -1.0], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1

   1. Initial program 99.9%

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

   if -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 59.3%

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

   2. Taylor expanded in im around 0 98.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 98.9%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right)} \]

      2. neg-mul-1 98.9%

         \[\leadsto \left(\color{blue}{\left(-im \cdot \sin re\right)} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      3. distribute-lft-neg-in 98.9%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      4. associate-*r* 98.9%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      5. *-commutative 98.9%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \cdot \sin re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      6. distribute-rgt-in 98.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      7. +-commutative 98.8%

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      8. sub-neg 98.8%

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      9. +-commutative 98.8%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      10. associate-*r* 98.8%

          \[\leadsto \left(\color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      11. associate-*r* 98.8%

          \[\leadsto \left(\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re + \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \sin re}\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      12. distribute-rgt-out 98.8%

          \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + -0.0001984126984126984 \cdot {im}^{7}\right)} + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \]

Alternative 7: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00105:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.45e+57)
     t_0
     (if (<= im 0.00105)
       (log1p (expm1 (* (- im) (sin re))))
       (if (<= im 1.1e+44) (* (- (exp (- im)) (exp im)) (* 0.5 re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.45e+57) {
		tmp = t_0;
	} else if (im <= 0.00105) {
		tmp = log1p(expm1((-im * sin(re))));
	} else if (im <= 1.1e+44) {
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.45e+57) {
		tmp = t_0;
	} else if (im <= 0.00105) {
		tmp = Math.log1p(Math.expm1((-im * Math.sin(re))));
	} else if (im <= 1.1e+44) {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.45e+57:
		tmp = t_0
	elif im <= 0.00105:
		tmp = math.log1p(math.expm1((-im * math.sin(re))))
	elif im <= 1.1e+44:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.45e+57)
		tmp = t_0;
	elseif (im <= 0.00105)
		tmp = log1p(expm1(Float64(Float64(-im) * sin(re))));
	elseif (im <= 1.1e+44)
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.45e+57], t$95$0, If[LessEqual[im, 0.00105], N[Log[1 + N[(Exp[N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.4500000000000001e57 or 1.09999999999999998e44 < 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 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -1.4500000000000001e57 < im < 0.00104999999999999994

   1. Initial program 41.0%

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

   2. Taylor expanded in im around 0 92.9%

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

      1. associate-*r* 92.9%

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

      2. neg-mul-1 92.9%

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

   4. Simplified 92.9%

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

      1. log1p-expm1-u 98.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)} \]

      2. *-commutative 98.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sin re \cdot \left(-im\right)}\right)\right) \]

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-im\right)\right)\right)} \]

   if 0.00104999999999999994 < im < 1.09999999999999998e44

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 84.6%

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

      1. associate-*r* 84.6%

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

      2. *-commutative 84.6%

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

   4. Simplified 84.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.45 \cdot 10^{+57}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq 0.00105:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 8: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0225:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -5.7)
     t_0
     (if (<= im 0.0225)
       (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (if (<= im 1.1e+44) (* (- (exp (- im)) (exp im)) (* 0.5 re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.0225) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	} 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 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-5.7d0)) then
        tmp = t_0
    else if (im <= 0.0225d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) then
        tmp = (exp(-im) - exp(im)) * (0.5d0 * re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.0225) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -5.7:
		tmp = t_0
	elif im <= 0.0225:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -5.7)
		tmp = t_0;
	elseif (im <= 0.0225)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -5.7)
		tmp = t_0;
	elseif (im <= 0.0225)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.7], t$95$0, If[LessEqual[im, 0.0225], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.70000000000000018 or 1.09999999999999998e44 < 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 98.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 98.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -5.70000000000000018 < im < 0.022499999999999999

   1. Initial program 37.1%

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

   2. Taylor expanded in im around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. associate-*r* 99.4%

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

      5. distribute-rgt-out-- 99.4%

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

      6. *-commutative 99.4%

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

   4. Simplified 99.4%

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

   if 0.022499999999999999 < im < 1.09999999999999998e44

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 84.6%

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

      1. associate-*r* 84.6%

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

      2. *-commutative 84.6%

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

   4. Simplified 84.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq 0.0225:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 9: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.7 \lor \neg \left(im \leq 5.6\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.7) (not (<= im 5.6)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.7) || !(im <= 5.6)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} 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) :: tmp
    if ((im <= (-5.7d0)) .or. (.not. (im <= 5.6d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    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 tmp;
	if ((im <= -5.7) || !(im <= 5.6)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.7) or not (im <= 5.6):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.7) || !(im <= 5.6))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.7) || ~((im <= 5.6)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.7], N[Not[LessEqual[im, 5.6]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $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 im < -5.70000000000000018 or 5.5999999999999996 < 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 85.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 85.7%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -5.70000000000000018 < im < 5.5999999999999996

   1. Initial program 37.7%

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

   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 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7 \lor \neg \left(im \leq 5.6\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 10: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.2) (not (<= im 4.2)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.2) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} 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 <= (-4.2d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.2) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.2) or not (im <= 4.2):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.2) || !(im <= 4.2))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.2) || ~((im <= 4.2)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.2], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.20000000000000018 or 4.20000000000000018 < 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 85.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 85.7%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -4.20000000000000018 < im < 4.20000000000000018

   1. Initial program 37.7%

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

   2. Taylor expanded in im around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 11: 82.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+46} \lor \neg \left(im \leq 30500000000\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.3e+46) (not (<= im 30500000000.0)))
   (* -0.0001984126984126984 (* re (pow im 7.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.3e+46) || !(im <= 30500000000.0)) {
		tmp = -0.0001984126984126984 * (re * pow(im, 7.0));
	} 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 <= (-1.3d+46)) .or. (.not. (im <= 30500000000.0d0))) then
        tmp = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.3e+46) || !(im <= 30500000000.0)) {
		tmp = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.3e+46) or not (im <= 30500000000.0):
		tmp = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.3e+46) || !(im <= 30500000000.0))
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.3e+46) || ~((im <= 30500000000.0)))
		tmp = -0.0001984126984126984 * (re * (im ^ 7.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.3e+46], N[Not[LessEqual[im, 30500000000.0]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.30000000000000007e46 or 3.05e10 < 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 91.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around 0 91.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 91.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right)} \]

      2. neg-mul-1 91.4%

         \[\leadsto \left(\color{blue}{\left(-im \cdot \sin re\right)} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      3. distribute-lft-neg-in 91.4%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      4. associate-*r* 91.4%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      5. *-commutative 91.4%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \cdot \sin re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      6. distribute-rgt-in 91.4%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      7. +-commutative 91.4%

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      8. sub-neg 91.4%

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) \]

      9. +-commutative 91.4%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      10. associate-*r* 91.4%

          \[\leadsto \left(\color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      11. associate-*r* 91.4%

          \[\leadsto \left(\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re + \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \sin re}\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      12. distribute-rgt-out 91.4%

          \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + -0.0001984126984126984 \cdot {im}^{7}\right)} + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   6. Taylor expanded in im around inf 91.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 91.4%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \sin re} \]

      2. *-commutative 91.4%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot -0.0001984126984126984\right)} \cdot \sin re \]

      3. associate-*l* 91.4%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \sin re\right)} \]

   8. Simplified 91.4%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \sin re\right)} \]

   9. Taylor expanded in re around 0 76.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot re\right)} \]

   if -1.30000000000000007e46 < im < 3.05e10

   1. Initial program 42.3%

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

   2. Taylor expanded in im around 0 91.2%

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

      1. associate-*r* 91.2%

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

      2. neg-mul-1 91.2%

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

   4. Simplified 91.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+46} \lor \neg \left(im \leq 30500000000\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 12: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.82 \cdot 10^{+177} \lor \neg \left(im \leq 340000000000\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.82e+177) (not (<= im 340000000000.0)))
   (* (- im) re)
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.82e+177) || !(im <= 340000000000.0)) {
		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 <= (-1.82d+177)) .or. (.not. (im <= 340000000000.0d0))) 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 <= -1.82e+177) || !(im <= 340000000000.0)) {
		tmp = -im * re;
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.82e+177) or not (im <= 340000000000.0):
		tmp = -im * re
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.82e+177) || !(im <= 340000000000.0))
		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 <= -1.82e+177) || ~((im <= 340000000000.0)))
		tmp = -im * re;
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.82e+177], N[Not[LessEqual[im, 340000000000.0]], $MachinePrecision]], N[((-im) * re), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.82e177 or 3.4e11 < 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 4.7%

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

      1. associate-*r* 4.7%

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

      2. neg-mul-1 4.7%

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

   4. Simplified 4.7%

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

   5. Taylor expanded in re around 0 15.7%

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

      1. associate-*r* 15.7%

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

      2. neg-mul-1 15.7%

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

   7. Simplified 15.7%

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

   if -1.82e177 < im < 3.4e11

   1. Initial program 54.2%

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

   2. Taylor expanded in im around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. neg-mul-1 73.0%

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

   4. Simplified 73.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.82 \cdot 10^{+177} \lor \neg \left(im \leq 340000000000\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 13: 32.5% 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 69.8%

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

2. Taylor expanded in im around 0 49.8%

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

   1. associate-*r* 49.8%

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

   2. neg-mul-1 49.8%

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

4. Simplified 49.8%

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

5. Taylor expanded in re around 0 32.9%

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

   1. associate-*r* 32.9%

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

   2. neg-mul-1 32.9%

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

7. Simplified 32.9%

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

8. Final simplification 32.9%

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

Developer target: 99.8% 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 2023293 
(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))))