math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 9

Speedup: 1.0×

• 49.4% 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: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460 \lor \neg \left(im \leq 2 \cdot 10^{+152}\right):\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 460.0) (not (<= im 2e+152)))
   (* (sin re) (+ (* 0.5 (* im im)) 1.0))
   (* (* 0.5 re) (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 460.0) || !(im <= 2e+152)) {
		tmp = sin(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 460.0d0) .or. (.not. (im <= 2d+152))) then
        tmp = sin(re) * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 460.0) || !(im <= 2e+152)) {
		tmp = Math.sin(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 460.0) or not (im <= 2e+152):
		tmp = math.sin(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 460.0) || !(im <= 2e+152))
		tmp = Float64(sin(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 460.0) || ~((im <= 2e+152)))
		tmp = sin(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 460.0], N[Not[LessEqual[im, 2e+152]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 460 or 2.0000000000000001e152 < 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 83.4%

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

   7. Simplified 83.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]
   8. Step-by-step derivation

      1. unpow2 45.1%

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

   9. Applied egg-rr 83.4%

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

   if 460 < im < 2.0000000000000001e152

   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 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460 \lor \neg \left(im \leq 2 \cdot 10^{+152}\right):\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 195000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 195000.0) (not (<= im 1.35e+154)))
   (* (sin re) (+ (* 0.5 (* im im)) 1.0))
   (sqrt (/ 0.0625 (pow re 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 195000.0) || !(im <= 1.35e+154)) {
		tmp = sin(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = sqrt((0.0625 / pow(re, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 195000.0d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = sin(re) * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = sqrt((0.0625d0 / (re ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 195000.0) || !(im <= 1.35e+154)) {
		tmp = Math.sin(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = Math.sqrt((0.0625 / Math.pow(re, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 195000.0) or not (im <= 1.35e+154):
		tmp = math.sin(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = math.sqrt((0.0625 / math.pow(re, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 195000.0) || !(im <= 1.35e+154))
		tmp = Float64(sin(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = sqrt(Float64(0.0625 / (re ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 195000.0) || ~((im <= 1.35e+154)))
		tmp = sin(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = sqrt((0.0625 / (re ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 195000.0], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0625 / N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 195000 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. 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%

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

   7. Simplified 83.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]
   8. Step-by-step derivation

      1. unpow2 45.1%

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

   9. Applied egg-rr 83.4%

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

   if 195000 < im < 1.35000000000000003e154

   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 19.2%

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

   7. Taylor expanded in re around 0 19.1%

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

      1. add-sqr-sqrt 19.1%

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

      2. sqrt-unprod 19.1%

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

      3. frac-times 19.1%

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

      4. metadata-eval 19.1%

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

      5. pow-prod-up 19.1%

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

      6. metadata-eval 19.1%

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

   9. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 195000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 195000.0) (not (<= im 1.35e+154)))
   (* (sin re) (+ (* 0.5 (* im im)) 1.0))
   (/ -0.25 (* re (- re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 195000.0) || !(im <= 1.35e+154)) {
		tmp = sin(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = -0.25 / (re * -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 <= 195000.0d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = sin(re) * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = (-0.25d0) / (re * -re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 195000.0) || !(im <= 1.35e+154)) {
		tmp = Math.sin(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = -0.25 / (re * -re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 195000.0) or not (im <= 1.35e+154):
		tmp = math.sin(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = -0.25 / (re * -re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 195000.0) || !(im <= 1.35e+154))
		tmp = Float64(sin(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(-0.25 / Float64(re * Float64(-re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 195000.0) || ~((im <= 1.35e+154)))
		tmp = sin(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = -0.25 / (re * -re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 195000.0], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-0.25 / N[(re * (-re)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 195000 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. 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%

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

   7. Simplified 83.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]
   8. Step-by-step derivation

      1. unpow2 45.1%

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

   9. Applied egg-rr 83.4%

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

   if 195000 < im < 1.35000000000000003e154

   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 19.2%

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

   7. Taylor expanded in re around 0 19.1%

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

      1. add-sqr-sqrt 19.1%

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

      2. sqrt-div 19.1%

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

      3. metadata-eval 19.1%

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

      4. sqrt-pow1 34.9%

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

      5. metadata-eval 34.9%

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

      6. pow1 34.9%

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

      7. sqrt-div 34.9%

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

      8. metadata-eval 34.9%

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

      9. sqrt-pow1 19.1%

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

      10. metadata-eval 19.1%

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

      11. pow1 19.1%

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

   9. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. frac-2neg 19.1%

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

       2. frac-times 19.1%

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

       3. metadata-eval 19.1%

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

       4. metadata-eval 19.1%

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

   11. Applied egg-rr 19.1%

       \[\leadsto \color{blue}{\frac{-0.25}{\left(-re\right) \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 195000.0)
   (sin re)
   (if (<= im 1e+154)
     (/ -0.25 (* re (- re)))
     (* re (+ (* 0.5 (* im im)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 195000.0) {
		tmp = sin(re);
	} else if (im <= 1e+154) {
		tmp = -0.25 / (re * -re);
	} else {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 195000.0d0) then
        tmp = sin(re)
    else if (im <= 1d+154) then
        tmp = (-0.25d0) / (re * -re)
    else
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 195000.0) {
		tmp = Math.sin(re);
	} else if (im <= 1e+154) {
		tmp = -0.25 / (re * -re);
	} else {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 195000.0:
		tmp = math.sin(re)
	elif im <= 1e+154:
		tmp = -0.25 / (re * -re)
	else:
		tmp = re * ((0.5 * (im * im)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 195000.0)
		tmp = sin(re);
	elseif (im <= 1e+154)
		tmp = Float64(-0.25 / Float64(re * Float64(-re)));
	else
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 195000.0)
		tmp = sin(re);
	elseif (im <= 1e+154)
		tmp = -0.25 / (re * -re);
	else
		tmp = re * ((0.5 * (im * im)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 195000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1e+154], N[(-0.25 / N[(re * (-re)), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 195000

   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 67.5%

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

   if 195000 < im < 1.00000000000000004e154

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. 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 19.2%

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

   7. Taylor expanded in re around 0 19.1%

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

      1. add-sqr-sqrt 19.1%

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

      2. sqrt-div 19.1%

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

      3. metadata-eval 19.1%

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

      4. sqrt-pow1 34.9%

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

      5. metadata-eval 34.9%

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

      6. pow1 34.9%

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

      7. sqrt-div 34.9%

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

      8. metadata-eval 34.9%

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

      9. sqrt-pow1 19.1%

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

      10. metadata-eval 19.1%

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

      11. pow1 19.1%

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

   9. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. frac-2neg 19.1%

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

       2. frac-times 19.1%

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

       3. metadata-eval 19.1%

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

       4. metadata-eval 19.1%

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

   11. Applied egg-rr 19.1%

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

   if 1.00000000000000004e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. 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 re around 0 72.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 72.0%

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

   10. Applied egg-rr 72.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.7% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 195000 \lor \neg \left(im \leq 1.75 \cdot 10^{+152}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 195000.0) (not (<= im 1.75e+152)))
   (* re (+ (* 0.5 (* im im)) 1.0))
   (/ -0.25 (* re (- re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 195000.0) || !(im <= 1.75e+152)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = -0.25 / (re * -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 <= 195000.0d0) .or. (.not. (im <= 1.75d+152))) then
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = (-0.25d0) / (re * -re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 195000.0) || !(im <= 1.75e+152)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = -0.25 / (re * -re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 195000.0) or not (im <= 1.75e+152):
		tmp = re * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = -0.25 / (re * -re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 195000.0) || !(im <= 1.75e+152))
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(-0.25 / Float64(re * Float64(-re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 195000.0) || ~((im <= 1.75e+152)))
		tmp = re * ((0.5 * (im * im)) + 1.0);
	else
		tmp = -0.25 / (re * -re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 195000.0], N[Not[LessEqual[im, 1.75e+152]], $MachinePrecision]], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-0.25 / N[(re * (-re)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 195000 or 1.74999999999999991e152 < 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 83.4%

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

   7. Simplified 83.4%

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

   8. Taylor expanded in re around 0 45.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 45.1%

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

   10. Applied egg-rr 45.1%

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

   if 195000 < im < 1.74999999999999991e152

   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 19.2%

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

   7. Taylor expanded in re around 0 19.1%

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

      1. add-sqr-sqrt 19.1%

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

      2. sqrt-div 19.1%

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

      3. metadata-eval 19.1%

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

      4. sqrt-pow1 34.9%

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

      5. metadata-eval 34.9%

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

      6. pow1 34.9%

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

      7. sqrt-div 34.9%

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

      8. metadata-eval 34.9%

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

      9. sqrt-pow1 19.1%

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

      10. metadata-eval 19.1%

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

      11. pow1 19.1%

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

   9. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. frac-2neg 19.1%

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

       2. frac-times 19.1%

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

       3. metadata-eval 19.1%

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

       4. metadata-eval 19.1%

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

   11. Applied egg-rr 19.1%

       \[\leadsto \color{blue}{\frac{-0.25}{\left(-re\right) \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000 \lor \neg \left(im \leq 1.75 \cdot 10^{+152}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 195000.0) re (/ -0.25 (* re (- re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 195000.0) {
		tmp = re;
	} else {
		tmp = -0.25 / (re * -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 <= 195000.0d0) then
        tmp = re
    else
        tmp = (-0.25d0) / (re * -re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 195000.0) {
		tmp = re;
	} else {
		tmp = -0.25 / (re * -re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 195000.0:
		tmp = re
	else:
		tmp = -0.25 / (re * -re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 195000.0)
		tmp = re;
	else
		tmp = Float64(-0.25 / Float64(re * Float64(-re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 195000.0)
		tmp = re;
	else
		tmp = -0.25 / (re * -re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 195000.0], re, N[(-0.25 / N[(re * (-re)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 195000

   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%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in re around 0 41.7%

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

   9. Taylor expanded in im around 0 30.3%

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

   if 195000 < 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 16.9%

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

   7. Taylor expanded in re around 0 16.7%

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

      1. add-sqr-sqrt 16.7%

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

      2. sqrt-div 16.7%

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

      3. metadata-eval 16.7%

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

      4. sqrt-pow1 37.8%

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

      5. metadata-eval 37.8%

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

      6. pow1 37.8%

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

      7. sqrt-div 37.8%

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

      8. metadata-eval 37.8%

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

      9. sqrt-pow1 16.7%

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

      10. metadata-eval 16.7%

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

      11. pow1 16.7%

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

   9. Applied egg-rr 16.7%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. frac-2neg 16.7%

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

       2. frac-times 16.7%

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

       3. metadata-eval 16.7%

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

       4. metadata-eval 16.7%

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

   11. Applied egg-rr 16.7%

       \[\leadsto \color{blue}{\frac{-0.25}{\left(-re\right) \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{re \cdot \left(-re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.6% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 195000.0) re (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 195000.0) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / 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 <= 195000.0d0) then
        tmp = re
    else
        tmp = (0.25d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 195000.0) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 195000.0:
		tmp = re
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 195000.0)
		tmp = re;
	else
		tmp = Float64(Float64(0.25 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 195000.0)
		tmp = re;
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 195000.0], re, N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 195000

   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%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in re around 0 41.7%

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

   9. Taylor expanded in im around 0 30.3%

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

   if 195000 < 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 16.9%

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

   7. Taylor expanded in re around 0 16.7%

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

      1. add-sqr-sqrt 16.7%

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

      2. sqrt-div 16.7%

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

      3. metadata-eval 16.7%

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

      4. sqrt-pow1 37.8%

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

      5. metadata-eval 37.8%

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

      6. pow1 37.8%

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

      7. sqrt-div 37.8%

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

      8. metadata-eval 37.8%

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

      9. sqrt-pow1 16.7%

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

      10. metadata-eval 16.7%

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

      11. pow1 16.7%

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

   9. Applied egg-rr 16.7%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. associate-*l/ 16.7%

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

       2. associate-*r/ 16.7%

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

       3. metadata-eval 16.7%

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

   11. Applied egg-rr 16.7%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

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 73.1%

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

7. Simplified 73.1%

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

8. Taylor expanded in re around 0 41.1%

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

9. Taylor expanded in im around 0 23.9%

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

10. Final simplification 23.9%

    \[\leadsto re \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))