math.cos on complex, imaginary part

Percentage Accurate: 66.3% → 99.6%

Time: 9.0s

Alternatives: 12

Speedup: 2.8×

• 49.8% 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: 66.3% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+53} \lor \neg \left(t_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (or (<= t_0 -1e+53) (not (<= t_0 2e-10)))
     (* 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 <= -1e+53) || !(t_0 <= 2e-10)) {
		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 <= (-1d+53)) .or. (.not. (t_0 <= 2d-10))) 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 <= -1e+53) || !(t_0 <= 2e-10)) {
		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 <= -1e+53) or not (t_0 <= 2e-10):
		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 <= -1e+53) || !(t_0 <= 2e-10))
		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 <= -1e+53) || ~((t_0 <= 2e-10)))
		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, -1e+53], N[Not[LessEqual[t$95$0, 2e-10]], $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)) < -9.9999999999999999e52 or 2.00000000000000007e-10 < (-.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 -9.9999999999999999e52 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.00000000000000007e-10

   1. Initial program 33.1%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+53} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.004 \lor \neg \left(t_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.004) (not (<= t_0 2e-10)))
     (* 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 <= -0.004) || !(t_0 <= 2e-10)) {
		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 <= (-0.004d0)) .or. (.not. (t_0 <= 2d-10))) 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 <= -0.004) || !(t_0 <= 2e-10)) {
		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 <= -0.004) or not (t_0 <= 2e-10):
		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 <= -0.004) || !(t_0 <= 2e-10))
		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 <= -0.004) || ~((t_0 <= 2e-10)))
		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, -0.004], N[Not[LessEqual[t$95$0, 2e-10]], $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)) < -0.0040000000000000001 or 2.00000000000000007e-10 < (-.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 -0.0040000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.00000000000000007e-10

   1. Initial program 32.3%

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

   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. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. 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%

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

   5. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.004 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -4.4 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.12:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 125:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* -0.008333333333333333 (* (sin re) (pow im 5.0)))))
   (if (<= im -4.4e+61)
     t_1
     (if (<= im -0.12)
       t_0
       (if (<= im 125.0)
         (* (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 = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	double tmp;
	if (im <= -4.4e+61) {
		tmp = t_1;
	} else if (im <= -0.12) {
		tmp = t_0;
	} else if (im <= 125.0) {
		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 = (-0.008333333333333333d0) * (sin(re) * (im ** 5.0d0))
    if (im <= (-4.4d+61)) then
        tmp = t_1
    else if (im <= (-0.12d0)) then
        tmp = t_0
    else if (im <= 125.0d0) 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 = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -4.4e+61) {
		tmp = t_1;
	} else if (im <= -0.12) {
		tmp = t_0;
	} else if (im <= 125.0) {
		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 = -0.008333333333333333 * (math.sin(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -4.4e+61:
		tmp = t_1
	elif im <= -0.12:
		tmp = t_0
	elif im <= 125.0:
		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(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -4.4e+61)
		tmp = t_1;
	elseif (im <= -0.12)
		tmp = t_0;
	elseif (im <= 125.0)
		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 = -0.008333333333333333 * (sin(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -4.4e+61)
		tmp = t_1;
	elseif (im <= -0.12)
		tmp = t_0;
	elseif (im <= 125.0)
		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[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.4e+61], t$95$1, If[LessEqual[im, -0.12], t$95$0, If[LessEqual[im, 125.0], 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 < -4.4000000000000001e61 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 0 100.0%

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

   3. Taylor expanded in im around inf 100.0%

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

   if -4.4000000000000001e61 < im < -0.12 or 125 < 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 84.6%

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

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

   if -0.12 < im < 125

   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 98.9%

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

   3. Taylor expanded in im around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. associate-*r* 98.5%

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

      5. distribute-rgt-out-- 98.5%

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

      6. *-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.4 \cdot 10^{+61}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.12:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 125:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 4: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -3.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 600000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (* (sin re) (pow im 5.0)))))
   (if (<= im -3.3)
     t_0
     (if (<= im 600000.0)
       (* im (- (sin re)))
       (if (<= im 7.5e+56)
         (* -0.008333333333333333 (* re (pow im 5.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	double tmp;
	if (im <= -3.3) {
		tmp = t_0;
	} else if (im <= 600000.0) {
		tmp = im * -sin(re);
	} else if (im <= 7.5e+56) {
		tmp = -0.008333333333333333 * (re * pow(im, 5.0));
	} 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) * (sin(re) * (im ** 5.0d0))
    if (im <= (-3.3d0)) then
        tmp = t_0
    else if (im <= 600000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 7.5d+56) then
        tmp = (-0.008333333333333333d0) * (re * (im ** 5.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -3.3) {
		tmp = t_0;
	} else if (im <= 600000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 7.5e+56) {
		tmp = -0.008333333333333333 * (re * Math.pow(im, 5.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.008333333333333333 * (math.sin(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -3.3:
		tmp = t_0
	elif im <= 600000.0:
		tmp = im * -math.sin(re)
	elif im <= 7.5e+56:
		tmp = -0.008333333333333333 * (re * math.pow(im, 5.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -3.3)
		tmp = t_0;
	elseif (im <= 600000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 7.5e+56)
		tmp = Float64(-0.008333333333333333 * Float64(re * (im ^ 5.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.008333333333333333 * (sin(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -3.3)
		tmp = t_0;
	elseif (im <= 600000.0)
		tmp = im * -sin(re);
	elseif (im <= 7.5e+56)
		tmp = -0.008333333333333333 * (re * (im ^ 5.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.3], t$95$0, If[LessEqual[im, 600000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 7.5e+56], N[(-0.008333333333333333 * N[(re * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.2999999999999998 or 7.4999999999999999e56 < 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%

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

   3. Taylor expanded in im around inf 85.8%

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

   if -3.2999999999999998 < im < 6e5

   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 97.8%

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

      1. associate-*r* 97.8%

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

      2. neg-mul-1 97.8%

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

   4. Simplified 97.8%

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

   if 6e5 < im < 7.4999999999999999e56

   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 4.2%

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

   3. Taylor expanded in im around inf 4.2%

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

   4. Taylor expanded in re around 0 51.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 600000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 5: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -4.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 115000:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (* (sin re) (pow im 5.0)))))
   (if (<= im -4.8)
     t_0
     (if (<= im 115000.0)
       (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (if (<= im 7.5e+56)
         (* -0.008333333333333333 (* re (pow im 5.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	double tmp;
	if (im <= -4.8) {
		tmp = t_0;
	} else if (im <= 115000.0) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 7.5e+56) {
		tmp = -0.008333333333333333 * (re * pow(im, 5.0));
	} 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) * (sin(re) * (im ** 5.0d0))
    if (im <= (-4.8d0)) then
        tmp = t_0
    else if (im <= 115000.0d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 7.5d+56) then
        tmp = (-0.008333333333333333d0) * (re * (im ** 5.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -4.8) {
		tmp = t_0;
	} else if (im <= 115000.0) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 7.5e+56) {
		tmp = -0.008333333333333333 * (re * Math.pow(im, 5.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.008333333333333333 * (math.sin(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -4.8:
		tmp = t_0
	elif im <= 115000.0:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 7.5e+56:
		tmp = -0.008333333333333333 * (re * math.pow(im, 5.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -4.8)
		tmp = t_0;
	elseif (im <= 115000.0)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 7.5e+56)
		tmp = Float64(-0.008333333333333333 * Float64(re * (im ^ 5.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.008333333333333333 * (sin(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -4.8)
		tmp = t_0;
	elseif (im <= 115000.0)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 7.5e+56)
		tmp = -0.008333333333333333 * (re * (im ^ 5.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.8], t$95$0, If[LessEqual[im, 115000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.5e+56], N[(-0.008333333333333333 * N[(re * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.79999999999999982 or 7.4999999999999999e56 < 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%

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

   3. Taylor expanded in im around inf 85.8%

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

   if -4.79999999999999982 < im < 115000

   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 98.9%

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

   3. Taylor expanded in im around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. associate-*r* 98.5%

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

      5. distribute-rgt-out-- 98.5%

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

      6. *-commutative 98.5%

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

   5. Simplified 98.5%

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

   if 115000 < im < 7.4999999999999999e56

   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 4.2%

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

   3. Taylor expanded in im around inf 4.2%

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

   4. Taylor expanded in re around 0 51.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.8:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 115000:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 6: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (- (* (pow im 5.0) -0.008333333333333333) im)))

C

double code(double re, double im) {
	return sin(re) * ((pow(im, 5.0) * -0.008333333333333333) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) - im);
}

Python

def code(re, im):
	return math.sin(re) * ((math.pow(im, 5.0) * -0.008333333333333333) - im)

Julia

function code(re, im)
	return Float64(sin(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (((im ^ 5.0) * -0.008333333333333333) - im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.5%

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

2. Taylor expanded in im around 0 89.7%

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

3. Taylor expanded in im around inf 89.1%

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

4. Taylor expanded in im around 0 89.1%

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

   1. +-commutative 89.1%

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

   2. associate-*r* 89.1%

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

   3. *-commutative 89.1%

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

   4. *-commutative 89.1%

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

   5. mul-1-neg 89.1%

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

   6. unsub-neg 89.1%

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

   7. *-commutative 89.1%

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

   8. distribute-rgt-out-- 89.1%

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

6. Simplified 89.1%

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

7. Final simplification 89.1%

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

Alternative 7: 81.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1750000 \lor \neg \left(im \leq 380\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1750000.0) (not (<= im 380.0)))
   (* -0.008333333333333333 (* re (pow im 5.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1750000.0) || !(im <= 380.0)) {
		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 <= (-1750000.0d0)) .or. (.not. (im <= 380.0d0))) 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 <= -1750000.0) || !(im <= 380.0)) {
		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 <= -1750000.0) or not (im <= 380.0):
		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 <= -1750000.0) || !(im <= 380.0))
		tmp = Float64(-0.008333333333333333 * Float64(re * (im ^ 5.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1750000.0) || ~((im <= 380.0)))
		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, -1750000.0], N[Not[LessEqual[im, 380.0]], $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 < -1.75e6 or 380 < 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 78.9%

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

   3. Taylor expanded in im around inf 78.9%

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

   4. Taylor expanded in re around 0 65.0%

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

   if -1.75e6 < im < 380

   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 97.8%

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

      1. associate-*r* 97.8%

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

      2. neg-mul-1 97.8%

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

   4. Simplified 97.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1750000 \lor \neg \left(im \leq 380\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.3 \cdot 10^{+21} \lor \neg \left(im \leq 75000000\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.3e+21) (not (<= im 75000000.0)))
   (* (- im) re)
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.3e+21) || !(im <= 75000000.0)) {
		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 <= (-4.3d+21)) .or. (.not. (im <= 75000000.0d0))) 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 <= -4.3e+21) || !(im <= 75000000.0)) {
		tmp = -im * re;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.3e+21) or not (im <= 75000000.0):
		tmp = -im * re
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.3e+21) || !(im <= 75000000.0))
		tmp = Float64(Float64(-im) * re);
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.3e+21) || ~((im <= 75000000.0)))
		tmp = -im * re;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.3e+21], N[Not[LessEqual[im, 75000000.0]], $MachinePrecision]], N[((-im) * re), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.3e21 or 7.5e7 < 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 4.5%

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

      1. associate-*r* 4.5%

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

      2. neg-mul-1 4.5%

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

   4. Simplified 4.5%

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

   5. Taylor expanded in re around 0 18.9%

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

      1. associate-*r* 18.9%

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

      2. neg-mul-1 18.9%

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

   7. Simplified 18.9%

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

   if -4.3e21 < im < 7.5e7

   1. Initial program 37.3%

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

   2. Taylor expanded in im around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

   4. Simplified 93.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.3 \cdot 10^{+21} \lor \neg \left(im \leq 75000000\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 9: 33.2% 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 64.5%

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

2. Taylor expanded in im around 0 54.7%

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

   1. associate-*r* 54.7%

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

   2. neg-mul-1 54.7%

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

4. Simplified 54.7%

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

5. Taylor expanded in re around 0 34.7%

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

   1. associate-*r* 34.7%

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

   2. neg-mul-1 34.7%

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

7. Simplified 34.7%

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

8. Final simplification 34.7%

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

Alternative 10: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -8.0)

C

double code(double re, double im) {
	return -8.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -8.0d0
end function

Java

public static double code(double re, double im) {
	return -8.0;
}

Python

def code(re, im):
	return -8.0

Julia

function code(re, im)
	return -8.0
end

MATLAB

function tmp = code(re, im)
	tmp = -8.0;
end

Wolfram

code[re_, im_] := -8.0

Derivation

1. Initial program 64.5%

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

2. Taylor expanded in im around 0 89.7%

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

4. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-8} \]

5. Final simplification 2.9%

   \[\leadsto -8 \]

Alternative 11: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.004629629629629629 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.004629629629629629)

C

double code(double re, double im) {
	return -0.004629629629629629;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -0.004629629629629629d0
end function

Java

public static double code(double re, double im) {
	return -0.004629629629629629;
}

Python

def code(re, im):
	return -0.004629629629629629

Julia

function code(re, im)
	return -0.004629629629629629
end

MATLAB

function tmp = code(re, im)
	tmp = -0.004629629629629629;
end

Wolfram

code[re_, im_] := -0.004629629629629629

Derivation

1. Initial program 64.5%

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

2. Taylor expanded in im around 0 89.7%

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

4. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{-0.004629629629629629} \]

5. Final simplification 3.0%

   \[\leadsto -0.004629629629629629 \]

Alternative 12: 15.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 64.5%

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

2. Taylor expanded in im around 0 89.7%

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

4. Applied egg-rr 17.0%

   \[\leadsto \color{blue}{0} \]

5. Final simplification 17.0%

   \[\leadsto 0 \]

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 2023318 
(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))))