math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 17

Speedup: 1.0×

• 49.3% 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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.1:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.1)
   (sin re)
   (if (<= im 1.32e+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 <= 7.1) {
		tmp = sin(re);
	} else if (im <= 1.32e+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 <= 7.1d0) then
        tmp = sin(re)
    else if (im <= 1.32d+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 <= 7.1) {
		tmp = Math.sin(re);
	} else if (im <= 1.32e+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 <= 7.1:
		tmp = math.sin(re)
	elif im <= 1.32e+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 <= 7.1)
		tmp = sin(re);
	elseif (im <= 1.32e+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 <= 7.1)
		tmp = sin(re);
	elseif (im <= 1.32e+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, 7.1], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.32e+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 < 7.0999999999999996

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 65.3%

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

   if 7.0999999999999996 < im < 1.31999999999999998e154

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 88.5%

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

   7. Simplified 88.5%

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

   if 1.31999999999999998e154 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.1:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 7.1:\\ \;\;\;\;\sin re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 7.1)
     (* (sin re) (+ t_0 1.0))
     (if (<= im 1.32e+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 <= 7.1) {
		tmp = sin(re) * (t_0 + 1.0);
	} else if (im <= 1.32e+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 <= 7.1d0) then
        tmp = sin(re) * (t_0 + 1.0d0)
    else if (im <= 1.32d+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 <= 7.1) {
		tmp = Math.sin(re) * (t_0 + 1.0);
	} else if (im <= 1.32e+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 <= 7.1:
		tmp = math.sin(re) * (t_0 + 1.0)
	elif im <= 1.32e+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 <= 7.1)
		tmp = Float64(sin(re) * Float64(t_0 + 1.0));
	elseif (im <= 1.32e+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 <= 7.1)
		tmp = sin(re) * (t_0 + 1.0);
	elseif (im <= 1.32e+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, 7.1], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.32e+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 < 7.0999999999999996

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 81.7%

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

   7. Simplified 81.7%

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

   if 7.0999999999999996 < im < 1.31999999999999998e154

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 88.5%

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

   7. Simplified 88.5%

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

   if 1.31999999999999998e154 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.1:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 760.0)
   (sin re)
   (if (<= im 4.2e+152)
     (cbrt (pow (/ 0.5 re) 6.0))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = sin(re);
	} else if (im <= 4.2e+152) {
		tmp = cbrt(pow((0.5 / re), 6.0));
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.2e+152) {
		tmp = Math.cbrt(Math.pow((0.5 / re), 6.0));
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 4.2e+152)
		tmp = cbrt((Float64(0.5 / re) ^ 6.0));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 760.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.2e+152], N[Power[N[Power[N[(0.5 / re), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $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 < 760

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 64.6%

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

   if 760 < im < 4.2000000000000003e152

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 26.7%

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

   7. Taylor expanded in re around 0 26.6%

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

      1. add-cbrt-cube 35.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}\right) \cdot \frac{0.25}{{re}^{2}}}} \]

      2. pow1/3 35.1%

         \[\leadsto \color{blue}{{\left(\left(\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}\right) \cdot \frac{0.25}{{re}^{2}}\right)}^{0.3333333333333333}} \]

      3. pow3 35.1%

         \[\leadsto {\color{blue}{\left({\left(\frac{0.25}{{re}^{2}}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. add-sqr-sqrt 35.1%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. pow2 35.1%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{\frac{0.25}{{re}^{2}}}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 35.1%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{\frac{0.25}{{re}^{2}}}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. sqrt-div 35.1%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 35.1%

         \[\leadsto {\left({\left(\frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. sqrt-pow1 35.1%

         \[\leadsto {\left({\left(\frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 35.1%

          \[\leadsto {\left({\left(\frac{0.5}{{re}^{\color{blue}{1}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. pow1 35.1%

          \[\leadsto {\left({\left(\frac{0.5}{\color{blue}{re}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. metadata-eval 35.1%

          \[\leadsto {\left({\left(\frac{0.5}{re}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{{\left({\left(\frac{0.5}{re}\right)}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 35.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   11. Simplified 35.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   if 4.2000000000000003e152 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in im around inf 97.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+129}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 850.0)
   (sin re)
   (if (<= im 3e+129)
     (cbrt (pow (/ 0.5 re) 6.0))
     (* (* 0.5 re) (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = sin(re);
	} else if (im <= 3e+129) {
		tmp = cbrt(pow((0.5 / re), 6.0));
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 850.0)
		tmp = sin(re);
	elseif (im <= 3e+129)
		tmp = cbrt((Float64(0.5 / re) ^ 6.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 850.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3e+129], N[Power[N[Power[N[(0.5 / re), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 850

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 64.6%

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

   if 850 < im < 3.0000000000000003e129

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 29.7%

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

   7. Taylor expanded in re around 0 29.7%

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

      1. add-cbrt-cube 40.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}\right) \cdot \frac{0.25}{{re}^{2}}}} \]

      2. pow1/3 40.6%

         \[\leadsto \color{blue}{{\left(\left(\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}\right) \cdot \frac{0.25}{{re}^{2}}\right)}^{0.3333333333333333}} \]

      3. pow3 40.6%

         \[\leadsto {\color{blue}{\left({\left(\frac{0.25}{{re}^{2}}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. add-sqr-sqrt 40.6%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. pow2 40.6%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{\frac{0.25}{{re}^{2}}}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 40.6%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{\frac{0.25}{{re}^{2}}}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. sqrt-div 40.6%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 40.6%

         \[\leadsto {\left({\left(\frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. sqrt-pow1 40.6%

         \[\leadsto {\left({\left(\frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 40.6%

          \[\leadsto {\left({\left(\frac{0.5}{{re}^{\color{blue}{1}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. pow1 40.6%

          \[\leadsto {\left({\left(\frac{0.5}{\color{blue}{re}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. metadata-eval 40.6%

          \[\leadsto {\left({\left(\frac{0.5}{re}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{0.5}{re}\right)}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 40.6%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   11. Simplified 40.6%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   if 3.0000000000000003e129 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in im around 0 68.1%

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

      1. +-commutative 68.1%

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

      2. unpow2 68.1%

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

      3. fma-define 68.1%

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

   10. Simplified 68.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+129}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 650.0)
   (sin re)
   (if (<= im 8.2e+128) (/ 0.25 (pow re 2.0)) (* (* 0.5 re) (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 650.0) {
		tmp = sin(re);
	} else if (im <= 8.2e+128) {
		tmp = 0.25 / pow(re, 2.0);
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 650.0)
		tmp = sin(re);
	elseif (im <= 8.2e+128)
		tmp = Float64(0.25 / (re ^ 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 650.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 8.2e+128], N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 650

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 64.6%

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

   if 650 < im < 8.20000000000000023e128

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 29.7%

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

   7. Taylor expanded in re around 0 29.7%

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

   if 8.20000000000000023e128 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in im around 0 68.1%

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

      1. +-commutative 68.1%

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

      2. unpow2 68.1%

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

      3. fma-define 68.1%

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

   10. Simplified 68.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 920:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+129}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 920.0)
   (sin re)
   (if (<= im 1.95e+129) (/ 0.25 (pow re 2.0)) (* (pow im 2.0) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 920.0) {
		tmp = sin(re);
	} else if (im <= 1.95e+129) {
		tmp = 0.25 / pow(re, 2.0);
	} else {
		tmp = pow(im, 2.0) * (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 <= 920.0d0) then
        tmp = sin(re)
    else if (im <= 1.95d+129) then
        tmp = 0.25d0 / (re ** 2.0d0)
    else
        tmp = (im ** 2.0d0) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 920.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.95e+129) {
		tmp = 0.25 / Math.pow(re, 2.0);
	} else {
		tmp = Math.pow(im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 920.0:
		tmp = math.sin(re)
	elif im <= 1.95e+129:
		tmp = 0.25 / math.pow(re, 2.0)
	else:
		tmp = math.pow(im, 2.0) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 920.0)
		tmp = sin(re);
	elseif (im <= 1.95e+129)
		tmp = Float64(0.25 / (re ^ 2.0));
	else
		tmp = Float64((im ^ 2.0) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 920.0)
		tmp = sin(re);
	elseif (im <= 1.95e+129)
		tmp = 0.25 / (re ^ 2.0);
	else
		tmp = (im ^ 2.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 920.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.95e+129], N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 2.0], $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 920

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 64.6%

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

   if 920 < im < 1.9499999999999999e129

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 29.7%

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

   7. Taylor expanded in re around 0 29.7%

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

   if 1.9499999999999999e129 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in im around 0 68.1%

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

      1. +-commutative 68.1%

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

      2. unpow2 68.1%

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

      3. fma-define 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in im around inf 68.1%

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

       1. *-commutative 68.1%

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

       2. associate-*l* 68.1%

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

       3. *-commutative 68.1%

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

   13. Simplified 68.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 920:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+129}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 920:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 920.0) (sin re) (/ 0.25 (pow re 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 920.0) {
		tmp = sin(re);
	} else {
		tmp = 0.25 / pow(re, 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 <= 920.0d0) then
        tmp = sin(re)
    else
        tmp = 0.25d0 / (re ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 920.0) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.25 / Math.pow(re, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 920.0:
		tmp = math.sin(re)
	else:
		tmp = 0.25 / math.pow(re, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 920.0)
		tmp = sin(re);
	else
		tmp = Float64(0.25 / (re ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 920.0)
		tmp = sin(re);
	else
		tmp = 0.25 / (re ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 920.0], N[Sin[re], $MachinePrecision], N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 920

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 64.6%

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

   if 920 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 21.0%

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

   7. Taylor expanded in re around 0 20.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 920:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 800.0) (sin re) (* (/ 0.5 re) (/ 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 800.0) {
		tmp = sin(re);
	} else {
		tmp = (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 <= 800.0d0) then
        tmp = sin(re)
    else
        tmp = (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 <= 800.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 800.0:
		tmp = math.sin(re)
	else:
		tmp = (0.5 / re) * (0.5 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 800.0)
		tmp = sin(re);
	else
		tmp = 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 <= 800.0)
		tmp = sin(re);
	else
		tmp = (0.5 / re) * (0.5 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 800.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 800

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 64.6%

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

   if 800 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 21.0%

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

   7. Taylor expanded in re around 0 20.8%

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

      1. add-sqr-sqrt 20.8%

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

      2. sqrt-div 20.8%

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

      3. metadata-eval 20.8%

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

      4. sqrt-pow1 32.6%

         \[\leadsto \frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. metadata-eval 32.6%

         \[\leadsto \frac{0.5}{{re}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. pow1 32.6%

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

      7. sqrt-div 32.6%

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

      8. metadata-eval 32.6%

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

      9. sqrt-pow1 20.8%

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

      10. metadata-eval 20.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{{re}^{\color{blue}{1}}} \]

      11. pow1 20.8%

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

   9. Applied egg-rr 20.8%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.1% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1100:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1100.0) re (* (/ 0.5 re) (/ 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1100.0) {
		tmp = re;
	} else {
		tmp = (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 <= 1100.0d0) then
        tmp = re
    else
        tmp = (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 <= 1100.0) {
		tmp = re;
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1100.0:
		tmp = re
	else:
		tmp = (0.5 / re) * (0.5 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1100.0)
		tmp = re;
	else
		tmp = 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 <= 1100.0)
		tmp = re;
	else
		tmp = (0.5 / re) * (0.5 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1100.0], re, N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1100

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in im around 0 34.2%

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

   if 1100 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 21.0%

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

   7. Taylor expanded in re around 0 20.8%

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

      1. add-sqr-sqrt 20.8%

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

      2. sqrt-div 20.8%

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

      3. metadata-eval 20.8%

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

      4. sqrt-pow1 32.6%

         \[\leadsto \frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. metadata-eval 32.6%

         \[\leadsto \frac{0.5}{{re}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. pow1 32.6%

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

      7. sqrt-div 32.6%

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

      8. metadata-eval 32.6%

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

      9. sqrt-pow1 20.8%

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

      10. metadata-eval 20.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{{re}^{\color{blue}{1}}} \]

      11. pow1 20.8%

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

   9. Applied egg-rr 20.8%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1100:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.7% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 68000000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 68000000000000.0) re 0.5))

C

double code(double re, double im) {
	double tmp;
	if (re <= 68000000000000.0) {
		tmp = re;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 68000000000000.0d0) then
        tmp = re
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 68000000000000.0) {
		tmp = re;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 68000000000000.0:
		tmp = re
	else:
		tmp = 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 68000000000000.0)
		tmp = re;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 68000000000000.0)
		tmp = re;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 68000000000000.0], re, 0.5]

Derivation

1. Split input into 2 regimes

2. if re < 6.8e13

   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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in im around 0 32.6%

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

   if 6.8e13 < 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. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   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 76.6%

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

   7. Simplified 76.6%

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

   9. Applied egg-rr 7.5%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 68000000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 4.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -4.0)

C

double code(double re, double im) {
	return -4.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -4.0d0
end function

Java

public static double code(double re, double im) {
	return -4.0;
}

Python

def code(re, im):
	return -4.0

Julia

function code(re, im)
	return -4.0
end

MATLAB

function tmp = code(re, im)
	tmp = -4.0;
end

Wolfram

code[re_, im_] := -4.0

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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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 73.2%

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

7. Simplified 73.2%

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

9. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{-4} \]

10. Final simplification 4.2%

    \[\leadsto -4 \]
11. Add Preprocessing

Alternative 13: 4.8% 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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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 73.2%

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

7. Simplified 73.2%

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

9. Applied egg-rr 4.9%

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

10. Final simplification 4.9%

    \[\leadsto -1 \]
11. Add Preprocessing

Alternative 14: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0625)

C

double code(double re, double im) {
	return 0.0625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0625d0
end function

Java

public static double code(double re, double im) {
	return 0.0625;
}

Python

def code(re, im):
	return 0.0625

Julia

function code(re, im)
	return 0.0625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0625;
end

Wolfram

code[re_, im_] := 0.0625

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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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 73.2%

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

7. Simplified 73.2%

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

9. Applied egg-rr 4.0%

   \[\leadsto \color{blue}{0.0625} \]

10. Final simplification 4.0%

    \[\leadsto 0.0625 \]
11. Add Preprocessing

Alternative 15: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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 73.2%

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

7. Simplified 73.2%

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

9. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{0.125} \]

10. Final simplification 4.1%

    \[\leadsto 0.125 \]
11. Add Preprocessing

Alternative 16: 4.4% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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 73.2%

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

7. Simplified 73.2%

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

9. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{0.25} \]

10. Final simplification 4.2%

    \[\leadsto 0.25 \]
11. Add Preprocessing

Alternative 17: 4.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

double code(double re, double im) {
	return 0.5;
}

Fortran

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

Java

public static double code(double re, double im) {
	return 0.5;
}

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5;
end

Wolfram

code[re_, im_] := 0.5

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. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

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 73.2%

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

7. Simplified 73.2%

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

9. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{0.5} \]

10. Final simplification 4.3%

    \[\leadsto 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024056 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))