math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 14

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \]
6. Add Preprocessing

Alternative 2: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.5e-9)
   (sin re)
   (if (<= im 1.35e+154)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.5e-9) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.5d-9) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = sin(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.5e-9) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.5e-9:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.5e-9)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.5e-9)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = sin(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.5e-9], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.4999999999999999e-9

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 3.4999999999999999e-9 < im < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. neg-sub0 99.9%

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

      6. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 83.8%

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

      1. associate-*r* 83.8%

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

   7. Simplified 83.8%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 0.06:\\ \;\;\;\;\sin re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 0.06)
     (* (sin re) (+ t_0 1.0))
     (if (<= im 1e+154)
       (* (+ (exp (- im)) (exp im)) (* 0.5 re))
       (* (sin re) t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 0.06) {
		tmp = sin(re) * (t_0 + 1.0);
	} else if (im <= 1e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = sin(re) * 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 = 0.5d0 * (im ** 2.0d0)
    if (im <= 0.06d0) then
        tmp = sin(re) * (t_0 + 1.0d0)
    else if (im <= 1d+154) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = sin(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 0.06) {
		tmp = Math.sin(re) * (t_0 + 1.0);
	} else if (im <= 1e+154) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 0.06:
		tmp = math.sin(re) * (t_0 + 1.0)
	elif im <= 1e+154:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = math.sin(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 0.06)
		tmp = Float64(sin(re) * Float64(t_0 + 1.0));
	elseif (im <= 1e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 0.06)
		tmp = sin(re) * (t_0 + 1.0);
	elseif (im <= 1e+154)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = sin(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.06], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.059999999999999998

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. distribute-rgt1-in 85.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   7. Simplified 85.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   if 0.059999999999999998 < im < 1.00000000000000004e154

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 83.3%

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

      1. associate-*r* 83.3%

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

   7. Simplified 83.3%

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

   if 1.00000000000000004e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.06:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+152}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550.0)
   (sin re)
   (if (<= im 6.5e+152)
     (pow (* (sin re) -2.0) -2.0)
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = sin(re);
	} else if (im <= 6.5e+152) {
		tmp = pow((sin(re) * -2.0), -2.0);
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 550.0d0) then
        tmp = sin(re)
    else if (im <= 6.5d+152) then
        tmp = (sin(re) * (-2.0d0)) ** (-2.0d0)
    else
        tmp = sin(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.5e+152) {
		tmp = Math.pow((Math.sin(re) * -2.0), -2.0);
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550.0:
		tmp = math.sin(re)
	elif im <= 6.5e+152:
		tmp = math.pow((math.sin(re) * -2.0), -2.0)
	else:
		tmp = math.sin(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 6.5e+152)
		tmp = Float64(sin(re) * -2.0) ^ -2.0;
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 6.5e+152)
		tmp = (sin(re) * -2.0) ^ -2.0;
	else
		tmp = sin(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.5e+152], N[Power[N[(N[Sin[re], $MachinePrecision] * -2.0), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 550

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 550 < im < 6.4999999999999997e152

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 31.5%

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

   if 6.4999999999999997e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+152}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+152}:\\ \;\;\;\;0.08333333333333333 + \sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3800.0)
   (sin re)
   (if (<= im 6.5e+152)
     (+ 0.08333333333333333 (sqrt (/ 0.0625 (pow re 4.0))))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = sin(re);
	} else if (im <= 6.5e+152) {
		tmp = 0.08333333333333333 + sqrt((0.0625 / pow(re, 4.0)));
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3800.0d0) then
        tmp = sin(re)
    else if (im <= 6.5d+152) then
        tmp = 0.08333333333333333d0 + sqrt((0.0625d0 / (re ** 4.0d0)))
    else
        tmp = sin(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.5e+152) {
		tmp = 0.08333333333333333 + Math.sqrt((0.0625 / Math.pow(re, 4.0)));
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3800.0:
		tmp = math.sin(re)
	elif im <= 6.5e+152:
		tmp = 0.08333333333333333 + math.sqrt((0.0625 / math.pow(re, 4.0)))
	else:
		tmp = math.sin(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3800.0)
		tmp = sin(re);
	elseif (im <= 6.5e+152)
		tmp = Float64(0.08333333333333333 + sqrt(Float64(0.0625 / (re ^ 4.0))));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3800.0)
		tmp = sin(re);
	elseif (im <= 6.5e+152)
		tmp = 0.08333333333333333 + sqrt((0.0625 / (re ^ 4.0)));
	else
		tmp = sin(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3800.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.5e+152], N[(0.08333333333333333 + N[Sqrt[N[(0.0625 / N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3800

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 3800 < im < 6.4999999999999997e152

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 31.5%

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

   7. Taylor expanded in re around 0 31.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 31.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 31.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 31.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 31.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-unprod 37.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}}} \]

       3. frac-times 37.8%

          \[\leadsto 0.08333333333333333 + \sqrt{\color{blue}{\frac{0.25 \cdot 0.25}{{re}^{2} \cdot {re}^{2}}}} \]

       4. metadata-eval 37.8%

          \[\leadsto 0.08333333333333333 + \sqrt{\frac{\color{blue}{0.0625}}{{re}^{2} \cdot {re}^{2}}} \]

       5. pow-prod-up 37.8%

          \[\leadsto 0.08333333333333333 + \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(2 + 2\right)}}}} \]

       6. metadata-eval 37.8%

          \[\leadsto 0.08333333333333333 + \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   11. Applied egg-rr 37.8%

       \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]

   if 6.4999999999999997e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+152}:\\ \;\;\;\;0.08333333333333333 + \sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+152}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 620.0)
   (sin re)
   (if (<= im 6.2e+152)
     (pow (* (sin re) -2.0) -2.0)
     (* re (+ (* 0.5 (pow im 2.0)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 620.0) {
		tmp = sin(re);
	} else if (im <= 6.2e+152) {
		tmp = pow((sin(re) * -2.0), -2.0);
	} else {
		tmp = re * ((0.5 * pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 620.0d0) then
        tmp = sin(re)
    else if (im <= 6.2d+152) then
        tmp = (sin(re) * (-2.0d0)) ** (-2.0d0)
    else
        tmp = re * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 620.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.2e+152) {
		tmp = Math.pow((Math.sin(re) * -2.0), -2.0);
	} else {
		tmp = re * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 620.0:
		tmp = math.sin(re)
	elif im <= 6.2e+152:
		tmp = math.pow((math.sin(re) * -2.0), -2.0)
	else:
		tmp = re * ((0.5 * math.pow(im, 2.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 620.0)
		tmp = sin(re);
	elseif (im <= 6.2e+152)
		tmp = Float64(sin(re) * -2.0) ^ -2.0;
	else
		tmp = Float64(re * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 620.0)
		tmp = sin(re);
	elseif (im <= 6.2e+152)
		tmp = (sin(re) * -2.0) ^ -2.0;
	else
		tmp = re * ((0.5 * (im ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 620.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.2e+152], N[Power[N[(N[Sin[re], $MachinePrecision] * -2.0), $MachinePrecision], -2.0], $MachinePrecision], N[(re * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 620

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 620 < im < 6.2e152

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 31.5%

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

   if 6.2e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 87.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+152}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3800.0)
   (sin re)
   (if (<= im 1.7e+151)
     (+ 0.08333333333333333 (* (/ 0.5 re) (/ 0.5 re)))
     (* re (+ (* 0.5 (pow im 2.0)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = sin(re);
	} else if (im <= 1.7e+151) {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	} else {
		tmp = re * ((0.5 * pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3800.0d0) then
        tmp = sin(re)
    else if (im <= 1.7d+151) then
        tmp = 0.08333333333333333d0 + ((0.5d0 / re) * (0.5d0 / re))
    else
        tmp = re * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.7e+151) {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	} else {
		tmp = re * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3800.0:
		tmp = math.sin(re)
	elif im <= 1.7e+151:
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re))
	else:
		tmp = re * ((0.5 * math.pow(im, 2.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3800.0)
		tmp = sin(re);
	elseif (im <= 1.7e+151)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.5 / re) * Float64(0.5 / re)));
	else
		tmp = Float64(re * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3800.0)
		tmp = sin(re);
	elseif (im <= 1.7e+151)
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	else
		tmp = re * ((0.5 * (im ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3800.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.7e+151], N[(0.08333333333333333 + N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3800

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 3800 < im < 1.7e151

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 31.5%

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

   7. Taylor expanded in re around 0 31.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 31.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 31.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 31.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 31.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 31.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 31.5%

          \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 31.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 31.0%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 31.0%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 31.5%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 31.5%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]

   if 1.7e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 87.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3800.0)
   (sin re)
   (if (<= im 2.6e+150)
     (+ 0.08333333333333333 (* (/ 0.5 re) (/ 0.5 re)))
     (* re (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = sin(re);
	} else if (im <= 2.6e+150) {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3800.0d0) then
        tmp = sin(re)
    else if (im <= 2.6d+150) then
        tmp = 0.08333333333333333d0 + ((0.5d0 / re) * (0.5d0 / re))
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.6e+150) {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3800.0:
		tmp = math.sin(re)
	elif im <= 2.6e+150:
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re))
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3800.0)
		tmp = sin(re);
	elseif (im <= 2.6e+150)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.5 / re) * Float64(0.5 / re)));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3800.0)
		tmp = sin(re);
	elseif (im <= 2.6e+150)
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3800.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.6e+150], N[(0.08333333333333333 + N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3800

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 3800 < im < 2.60000000000000006e150

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 31.5%

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

   7. Taylor expanded in re around 0 31.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 31.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 31.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 31.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 31.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 31.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 31.5%

          \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 31.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 31.0%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 35.5%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 31.0%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 31.5%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 31.5%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]

   if 2.60000000000000006e150 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 87.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   7. Taylor expanded in im around 0 87.1%

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

   8. Taylor expanded in im around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*r* 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-*r* 87.1%

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

   10. Simplified 87.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3800.0)
   (sin re)
   (+ 0.08333333333333333 (* (/ 0.5 re) (/ 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = sin(re);
	} else {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / 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 <= 3800.0d0) then
        tmp = sin(re)
    else
        tmp = 0.08333333333333333d0 + ((0.5d0 / re) * (0.5d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3800.0:
		tmp = math.sin(re)
	else:
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3800.0)
		tmp = sin(re);
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.5 / re) * Float64(0.5 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3800.0)
		tmp = sin(re);
	else
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3800.0], N[Sin[re], $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3800

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 3800 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in re around 0 24.8%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 24.8%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 24.8%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 24.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 24.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 24.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 24.8%

          \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 24.8%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 24.4%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 24.4%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 24.8%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 24.8%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.3% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3800.0) re (+ 0.08333333333333333 (* (/ 0.5 re) (/ 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / 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 <= 3800.0d0) then
        tmp = re
    else
        tmp = 0.08333333333333333d0 + ((0.5d0 / re) * (0.5d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3800.0) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3800.0:
		tmp = re
	else:
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3800.0)
		tmp = re;
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.5 / re) * Float64(0.5 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3800.0)
		tmp = re;
	else
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3800.0], re, N[(0.08333333333333333 + N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3800

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.3%

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

   6. Taylor expanded in re around 0 48.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   7. Taylor expanded in im around 0 37.0%

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

   if 3800 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in re around 0 24.8%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 24.8%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 24.8%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 24.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 24.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 24.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 24.8%

          \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 24.8%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 24.4%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 32.1%

          \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 24.4%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 24.8%

           \[\leadsto 0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 24.8%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.7% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.0) re 1.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = re;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.0d0) then
        tmp = re
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = re;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.0:
		tmp = re
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.0)
		tmp = re;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.0)
		tmp = re;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.0], re, 1.0]

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 75.1%

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

   6. Taylor expanded in re around 0 57.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   7. Taylor expanded in im around 0 37.1%

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

   if 1 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 8.1%

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

      1. +-inverses 8.1%

         \[\leadsto \frac{\sin re \cdot -2}{\sin re \cdot -2 + \color{blue}{0}} \]

      2. +-rgt-identity 8.1%

         \[\leadsto \frac{\sin re \cdot -2}{\color{blue}{\sin re \cdot -2}} \]

      3. *-inverses 8.1%

         \[\leadsto \color{blue}{1} \]

   8. Simplified 8.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 2.9% accurate, 309.0× 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 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{\log \left({1}^{\sin re}\right)} \]
7. Step-by-step derivation

   1. pow-base-1 3.0%

      \[\leadsto \log \color{blue}{1} \]

   2. metadata-eval 3.0%

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

8. Simplified 3.0%

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

9. Final simplification 3.0%

   \[\leadsto 0 \]
10. Add Preprocessing

Alternative 13: 4.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 12.5%

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

7. Taylor expanded in re around 0 12.3%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
8. Step-by-step derivation

   1. associate-*r/ 12.3%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 12.3%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

9. Simplified 12.3%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

10. Taylor expanded in re around inf 4.3%

    \[\leadsto \color{blue}{0.08333333333333333} \]

11. Final simplification 4.3%

    \[\leadsto 0.08333333333333333 \]
12. Add Preprocessing

Alternative 14: 4.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 4.9%

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

   1. +-inverses 4.9%

      \[\leadsto \frac{\sin re \cdot -2}{\sin re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 4.9%

      \[\leadsto \frac{\sin re \cdot -2}{\color{blue}{\sin re \cdot -2}} \]

   3. *-inverses 4.9%

      \[\leadsto \color{blue}{1} \]

8. Simplified 4.9%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 4.9%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))