math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 13

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: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0064:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0064)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 1.4e+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 <= 0.0064) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 1.4e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0064)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 1.4e+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

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0064], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+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 < 0.00640000000000000031

   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 83.4%

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

      1. +-commutative 42.4%

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

      2. unpow2 42.4%

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

      3. fma-define 42.4%

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

   7. Simplified 83.4%

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

   if 0.00640000000000000031 < im < 1.4e154

   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 73.4%

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

   if 1.4e154 < 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%

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

      1. +-commutative 80.6%

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

      2. unpow2 80.6%

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

      3. fma-define 80.6%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   9. 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%

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

   10. 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 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0064:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{{re}^{-4}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16)
   (sin re)
   (if (<= im 3.8e+152)
     (sqrt (pow re -4.0))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = sin(re);
	} else if (im <= 3.8e+152) {
		tmp = sqrt(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 <= 4.9d+16) then
        tmp = sin(re)
    else if (im <= 3.8d+152) then
        tmp = sqrt((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 <= 4.9e+16) {
		tmp = Math.sin(re);
	} else if (im <= 3.8e+152) {
		tmp = Math.sqrt(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 <= 4.9e+16:
		tmp = math.sin(re)
	elif im <= 3.8e+152:
		tmp = math.sqrt(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 <= 4.9e+16)
		tmp = sin(re);
	elseif (im <= 3.8e+152)
		tmp = sqrt((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 <= 4.9e+16)
		tmp = sin(re);
	elseif (im <= 3.8e+152)
		tmp = sqrt((re ^ -4.0));
	else
		tmp = sin(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.8e+152], N[Sqrt[N[Power[re, -4.0], $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 < 4.9e16

   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 68.4%

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

   if 4.9e16 < im < 3.8e152

   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 2.5%

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

   7. Applied egg-rr 17.3%

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

   8. Taylor expanded in re around 0 17.1%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 17.1%

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

      2. sqrt-unprod 24.5%

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

      3. pow-flip 24.5%

         \[\leadsto \sqrt{\color{blue}{{re}^{\left(-2\right)}} \cdot \frac{1}{{re}^{2}}} \]

      4. metadata-eval 24.5%

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

      5. pow-flip 24.5%

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

      6. metadata-eval 24.5%

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

      7. pow-prod-up 24.5%

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

      8. metadata-eval 24.5%

         \[\leadsto \sqrt{{re}^{\color{blue}{-4}}} \]

   10. Applied egg-rr 24.5%

       \[\leadsto \color{blue}{\sqrt{{re}^{-4}}} \]

   if 3.8e152 < 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%

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

      1. +-commutative 78.4%

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

      2. unpow2 78.4%

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

      3. fma-define 78.4%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in im around inf 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{{re}^{-4}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{{re}^{-4}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 3.8e+152)
     (sqrt (pow re -4.0))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 3.8e+152) {
		tmp = sqrt(pow(re, -4.0));
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+16)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 3.8e+152)
		tmp = sqrt((re ^ -4.0));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.8e+152], N[Sqrt[N[Power[re, -4.0], $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 < 4.9e16

   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 81.3%

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

      1. +-commutative 41.5%

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

      2. unpow2 41.5%

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

      3. fma-define 41.5%

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

   7. Simplified 81.3%

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

   if 4.9e16 < im < 3.8e152

   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 2.5%

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

   7. Applied egg-rr 17.3%

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

   8. Taylor expanded in re around 0 17.1%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 17.1%

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

      2. sqrt-unprod 24.5%

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

      3. pow-flip 24.5%

         \[\leadsto \sqrt{\color{blue}{{re}^{\left(-2\right)}} \cdot \frac{1}{{re}^{2}}} \]

      4. metadata-eval 24.5%

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

      5. pow-flip 24.5%

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

      6. metadata-eval 24.5%

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

      7. pow-prod-up 24.5%

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

      8. metadata-eval 24.5%

         \[\leadsto \sqrt{{re}^{\color{blue}{-4}}} \]

   10. Applied egg-rr 24.5%

       \[\leadsto \color{blue}{\sqrt{{re}^{-4}}} \]

   if 3.8e152 < 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%

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

      1. +-commutative 78.4%

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

      2. unpow2 78.4%

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

      3. fma-define 78.4%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in im around inf 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{{re}^{-4}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 35:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 35.0)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 1.4e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 35.0) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 1.4e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 35.0)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 1.4e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 35.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $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 < 35

   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 83.0%

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

      1. +-commutative 42.3%

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

      2. unpow2 42.3%

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

      3. fma-define 42.3%

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

   7. Simplified 83.0%

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

   if 35 < im < 1.4e154

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   8. Applied egg-rr 51.2%

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

      1. +-inverses 51.2%

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

      2. +-rgt-identity 51.2%

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

      3. *-inverses 51.2%

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

   10. Simplified 51.2%

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

   if 1.4e154 < 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%

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

      1. +-commutative 80.6%

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

      2. unpow2 80.6%

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

      3. fma-define 80.6%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   9. 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%

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

   10. 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 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 35:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{{re}^{-4}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16)
   (sin re)
   (if (<= im 2.75e+132) (sqrt (pow re -4.0)) (* 0.5 (* re (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = sin(re);
	} else if (im <= 2.75e+132) {
		tmp = sqrt(pow(re, -4.0));
	} else {
		tmp = 0.5 * (re * 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 <= 4.9d+16) then
        tmp = sin(re)
    else if (im <= 2.75d+132) then
        tmp = sqrt((re ** (-4.0d0)))
    else
        tmp = 0.5d0 * (re * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = Math.sin(re);
	} else if (im <= 2.75e+132) {
		tmp = Math.sqrt(Math.pow(re, -4.0));
	} else {
		tmp = 0.5 * (re * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+16:
		tmp = math.sin(re)
	elif im <= 2.75e+132:
		tmp = math.sqrt(math.pow(re, -4.0))
	else:
		tmp = 0.5 * (re * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+16)
		tmp = sin(re);
	elseif (im <= 2.75e+132)
		tmp = sqrt((re ^ -4.0));
	else
		tmp = Float64(0.5 * Float64(re * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+16)
		tmp = sin(re);
	elseif (im <= 2.75e+132)
		tmp = sqrt((re ^ -4.0));
	else
		tmp = 0.5 * (re * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.75e+132], N[Sqrt[N[Power[re, -4.0], $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.9e16

   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 68.4%

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

   if 4.9e16 < im < 2.75e132

   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 2.5%

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

   7. Applied egg-rr 18.5%

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

   8. Taylor expanded in re around 0 18.3%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 18.3%

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

      2. sqrt-unprod 23.6%

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

      3. pow-flip 23.6%

         \[\leadsto \sqrt{\color{blue}{{re}^{\left(-2\right)}} \cdot \frac{1}{{re}^{2}}} \]

      4. metadata-eval 23.6%

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

      5. pow-flip 23.6%

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

      6. metadata-eval 23.6%

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

      7. pow-prod-up 23.6%

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

      8. metadata-eval 23.6%

         \[\leadsto \sqrt{{re}^{\color{blue}{-4}}} \]

   10. Applied egg-rr 23.6%

       \[\leadsto \color{blue}{\sqrt{{re}^{-4}}} \]

   if 2.75e132 < 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 80.0%

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

   6. Taylor expanded in im around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. fma-define 74.7%

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

   8. Simplified 74.7%

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

   9. Taylor expanded in im around inf 74.7%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{{re}^{-4}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16)
   (sin re)
   (if (<= im 1.06e+132) (/ 1.0 (pow re 2.0)) (* 0.5 (* re (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = sin(re);
	} else if (im <= 1.06e+132) {
		tmp = 1.0 / pow(re, 2.0);
	} else {
		tmp = 0.5 * (re * 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 <= 4.9d+16) then
        tmp = sin(re)
    else if (im <= 1.06d+132) then
        tmp = 1.0d0 / (re ** 2.0d0)
    else
        tmp = 0.5d0 * (re * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = Math.sin(re);
	} else if (im <= 1.06e+132) {
		tmp = 1.0 / Math.pow(re, 2.0);
	} else {
		tmp = 0.5 * (re * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+16:
		tmp = math.sin(re)
	elif im <= 1.06e+132:
		tmp = 1.0 / math.pow(re, 2.0)
	else:
		tmp = 0.5 * (re * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+16)
		tmp = sin(re);
	elseif (im <= 1.06e+132)
		tmp = Float64(1.0 / (re ^ 2.0));
	else
		tmp = Float64(0.5 * Float64(re * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+16)
		tmp = sin(re);
	elseif (im <= 1.06e+132)
		tmp = 1.0 / (re ^ 2.0);
	else
		tmp = 0.5 * (re * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.06e+132], N[(1.0 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.9e16

   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 68.4%

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

   if 4.9e16 < im < 1.0599999999999999e132

   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 2.5%

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

   7. Applied egg-rr 18.5%

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

   8. Taylor expanded in re around 0 18.3%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]

   if 1.0599999999999999e132 < 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 80.0%

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

   6. Taylor expanded in im around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. fma-define 74.7%

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

   8. Simplified 74.7%

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

   9. Taylor expanded in im around inf 74.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{re}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16) (sin re) (/ 1.0 (pow re 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = sin(re);
	} else {
		tmp = 1.0 / 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 <= 4.9d+16) then
        tmp = sin(re)
    else
        tmp = 1.0d0 / (re ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = Math.sin(re);
	} else {
		tmp = 1.0 / Math.pow(re, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+16:
		tmp = math.sin(re)
	else:
		tmp = 1.0 / math.pow(re, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+16)
		tmp = sin(re);
	else
		tmp = Float64(1.0 / (re ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+16)
		tmp = sin(re);
	else
		tmp = 1.0 / (re ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], N[Sin[re], $MachinePrecision], N[(1.0 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.9e16

   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 68.4%

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

   if 4.9e16 < 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 2.6%

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

   7. Applied egg-rr 17.4%

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

   8. Taylor expanded in re around 0 17.2%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{re} \cdot \frac{1}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16) (sin re) (* (/ 1.0 re) (/ 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = sin(re);
	} else {
		tmp = (1.0 / re) * (1.0 / re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.9d+16) then
        tmp = sin(re)
    else
        tmp = (1.0d0 / re) * (1.0d0 / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = Math.sin(re);
	} else {
		tmp = (1.0 / re) * (1.0 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+16:
		tmp = math.sin(re)
	else:
		tmp = (1.0 / re) * (1.0 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+16)
		tmp = sin(re);
	else
		tmp = Float64(Float64(1.0 / re) * Float64(1.0 / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+16)
		tmp = sin(re);
	else
		tmp = (1.0 / re) * (1.0 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], N[Sin[re], $MachinePrecision], N[(N[(1.0 / re), $MachinePrecision] * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.9e16

   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 68.4%

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

   if 4.9e16 < 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 2.6%

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

   7. Applied egg-rr 17.4%

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

   8. Taylor expanded in re around 0 17.2%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 17.2%

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

      2. sqrt-div 17.2%

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

      3. metadata-eval 17.2%

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

      4. sqrt-pow1 34.8%

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

      5. metadata-eval 34.8%

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

      6. pow1 34.8%

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

      7. sqrt-div 34.8%

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

      8. metadata-eval 34.8%

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

      9. sqrt-pow1 17.2%

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

      10. metadata-eval 17.2%

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

      11. pow1 17.2%

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

   10. Applied egg-rr 17.2%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{re} \cdot \frac{1}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.5% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{re} \cdot \frac{1}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+16) re (* (/ 1.0 re) (/ 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = re;
	} else {
		tmp = (1.0 / re) * (1.0 / re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.9d+16) then
        tmp = re
    else
        tmp = (1.0d0 / re) * (1.0d0 / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+16) {
		tmp = re;
	} else {
		tmp = (1.0 / re) * (1.0 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+16:
		tmp = re
	else:
		tmp = (1.0 / re) * (1.0 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+16)
		tmp = re;
	else
		tmp = Float64(Float64(1.0 / re) * Float64(1.0 / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+16)
		tmp = re;
	else
		tmp = (1.0 / re) * (1.0 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+16], re, N[(N[(1.0 / re), $MachinePrecision] * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.9e16

   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 68.4%

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

   6. Taylor expanded in re around 0 30.6%

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

   if 4.9e16 < 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 2.6%

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

   7. Applied egg-rr 17.4%

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

   8. Taylor expanded in re around 0 17.2%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 17.2%

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

      2. sqrt-div 17.2%

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

      3. metadata-eval 17.2%

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

      4. sqrt-pow1 34.8%

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

      5. metadata-eval 34.8%

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

      6. pow1 34.8%

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

      7. sqrt-div 34.8%

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

      8. metadata-eval 34.8%

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

      9. sqrt-pow1 17.2%

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

      10. metadata-eval 17.2%

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

      11. pow1 17.2%

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

   10. Applied egg-rr 17.2%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+16}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{re} \cdot \frac{1}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.6% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 5.4e-11) re 1.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.4e-11) {
		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 <= 5.4d-11) 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 <= 5.4e-11) {
		tmp = re;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.4e-11:
		tmp = re
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.4e-11)
		tmp = re;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.4e-11)
		tmp = re;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.4e-11], re, 1.0]

Derivation

1. Split input into 2 regimes

2. if re < 5.40000000000000009e-11

   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 47.2%

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

   6. Taylor expanded in re around 0 30.0%

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

   if 5.40000000000000009e-11 < 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

   6. Applied egg-rr 7.9%

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

      1. +-inverses 7.9%

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

      2. +-rgt-identity 7.9%

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

      3. *-inverses 7.9%

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

   8. Simplified 7.9%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
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. 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 2.9%

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

   1. pow-base-1 2.9%

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

   2. metadata-eval 2.9%

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

8. Simplified 2.9%

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

9. Final simplification 2.9%

   \[\leadsto 0 \]
10. 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

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 2024072 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))