math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

Alternatives: 15

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-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.06 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.06) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (* (+ (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.06) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 0.06d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.06) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.06) or not (im <= 1.35e+154):
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	else:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.06) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.06) || ~((im <= 1.35e+154)))
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	else
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.06], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.059999999999999998 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.8%

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

   6. Simplified 84.8%

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

   if 0.059999999999999998 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.06 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.036:\\ \;\;\;\;\sin re + \sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.036)
   (+ (sin re) (* (sin re) (* 0.5 (* im im))))
   (if (<= im 1.35e+154)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (* 0.5 (sin re)) (+ (* im im) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.036) {
		tmp = sin(re) + (sin(re) * (0.5 * (im * im)));
	} else if (im <= 1.35e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * ((im * 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 <= 0.036d0) then
        tmp = sin(re) + (sin(re) * (0.5d0 * (im * im)))
    else if (im <= 1.35d+154) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.036) {
		tmp = Math.sin(re) + (Math.sin(re) * (0.5 * (im * im)));
	} else if (im <= 1.35e+154) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.036:
		tmp = math.sin(re) + (math.sin(re) * (0.5 * (im * im)))
	elif im <= 1.35e+154:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.036)
		tmp = Float64(sin(re) + Float64(sin(re) * Float64(0.5 * Float64(im * im))));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.036)
		tmp = sin(re) + (sin(re) * (0.5 * (im * im)));
	elseif (im <= 1.35e+154)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.036], N[(N[Sin[re], $MachinePrecision] + N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0359999999999999973

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 0.0359999999999999973 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 74.3%

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

   6. Simplified 74.3%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. 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. Simplified 100.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.036:\\ \;\;\;\;\sin re + \sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]

Alternative 4: 81.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0145 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.0145) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (+ re (* re (+ (* 0.5 (* im im)) (* 0.041666666666666664 (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.0145) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = re + (re * ((0.5 * (im * im)) + (0.041666666666666664 * pow(im, 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 <= 0.0145d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else
        tmp = re + (re * ((0.5d0 * (im * im)) + (0.041666666666666664d0 * (im ** 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.0145) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = re + (re * ((0.5 * (im * im)) + (0.041666666666666664 * Math.pow(im, 4.0))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.0145) or not (im <= 1.35e+154):
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	else:
		tmp = re + (re * ((0.5 * (im * im)) + (0.041666666666666664 * math.pow(im, 4.0))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.0145) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(re + Float64(re * Float64(Float64(0.5 * Float64(im * im)) + Float64(0.041666666666666664 * (im ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.0145) || ~((im <= 1.35e+154)))
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	else
		tmp = re + (re * ((0.5 * (im * im)) + (0.041666666666666664 * (im ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.0145], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0145000000000000007 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.8%

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

   6. Simplified 84.8%

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

   if 0.0145000000000000007 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in im around 0 43.7%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 43.7%

         \[\leadsto re + \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re} + 0.5 \cdot \left({im}^{2} \cdot re\right)\right) \]

      2. associate-*r* 43.7%

         \[\leadsto re + \left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re}\right) \]

      3. distribute-rgt-out 43.7%

         \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right)} \]

      4. +-commutative 43.7%

         \[\leadsto re + re \cdot \color{blue}{\left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      5. unpow2 43.7%

         \[\leadsto re + re \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 0.041666666666666664 \cdot {im}^{4}\right) \]

   9. Simplified 43.7%

      \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 0.041666666666666664 \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0145 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+33} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 4.6e+33) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (+ re (* re (* 0.041666666666666664 (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 4.6e+33) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = re + (re * (0.041666666666666664 * pow(im, 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 <= 4.6d+33) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else
        tmp = re + (re * (0.041666666666666664d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 4.6e+33) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = re + (re * (0.041666666666666664 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 4.6e+33) or not (im <= 1.35e+154):
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	else:
		tmp = re + (re * (0.041666666666666664 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 4.6e+33) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(re + Float64(re * Float64(0.041666666666666664 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 4.6e+33) || ~((im <= 1.35e+154)))
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	else
		tmp = re + (re * (0.041666666666666664 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 4.6e+33], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.60000000000000021e33 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 81.6%

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

   6. Simplified 81.6%

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

   if 4.60000000000000021e33 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in im around 0 56.9%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 56.9%

         \[\leadsto re + \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re} + 0.5 \cdot \left({im}^{2} \cdot re\right)\right) \]

      2. associate-*r* 56.9%

         \[\leadsto re + \left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re}\right) \]

      3. distribute-rgt-out 56.9%

         \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right)} \]

      4. +-commutative 56.9%

         \[\leadsto re + re \cdot \color{blue}{\left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      5. unpow2 56.9%

         \[\leadsto re + re \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 0.041666666666666664 \cdot {im}^{4}\right) \]

   9. Simplified 56.9%

      \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   10. Taylor expanded in im around inf 56.9%

       \[\leadsto re + \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 56.9%

          \[\leadsto re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re} \]

       2. *-commutative 56.9%

          \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]

   12. Simplified 56.9%

       \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+33} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 2.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8e+30)
   (sin re)
   (* re (+ (* 0.041666666666666664 (pow im 4.0)) 2.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+30) {
		tmp = sin(re);
	} else {
		tmp = re * ((0.041666666666666664 * pow(im, 4.0)) + 2.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.8d+30) then
        tmp = sin(re)
    else
        tmp = re * ((0.041666666666666664d0 * (im ** 4.0d0)) + 2.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+30) {
		tmp = Math.sin(re);
	} else {
		tmp = re * ((0.041666666666666664 * Math.pow(im, 4.0)) + 2.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8e+30:
		tmp = math.sin(re)
	else:
		tmp = re * ((0.041666666666666664 * math.pow(im, 4.0)) + 2.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e+30)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(Float64(0.041666666666666664 * (im ^ 4.0)) + 2.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8e+30)
		tmp = sin(re);
	else
		tmp = re * ((0.041666666666666664 * (im ^ 4.0)) + 2.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e+30], N[Sin[re], $MachinePrecision], N[(re * N[(N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.8000000000000005e30

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.9%

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

   if 6.8000000000000005e30 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in im around 0 73.9%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 73.9%

         \[\leadsto re + \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re} + 0.5 \cdot \left({im}^{2} \cdot re\right)\right) \]

      2. associate-*r* 73.9%

         \[\leadsto re + \left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re}\right) \]

      3. distribute-rgt-out 73.9%

         \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right)} \]

      4. +-commutative 73.9%

         \[\leadsto re + re \cdot \color{blue}{\left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      5. unpow2 73.9%

         \[\leadsto re + re \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 0.041666666666666664 \cdot {im}^{4}\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   11. Applied egg-rr 73.9%

       \[\leadsto re + re \cdot \left(\color{blue}{1.5} + 0.041666666666666664 \cdot {im}^{4}\right) \]

   12. Taylor expanded in re around 0 73.9%

       \[\leadsto \color{blue}{re \cdot \left(2.5 + 0.041666666666666664 \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 2.5\right)\\ \end{array} \]

Alternative 7: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8e+30)
   (sin re)
   (+ re (* re (* 0.041666666666666664 (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+30) {
		tmp = sin(re);
	} else {
		tmp = re + (re * (0.041666666666666664 * pow(im, 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 <= 6.8d+30) then
        tmp = sin(re)
    else
        tmp = re + (re * (0.041666666666666664d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+30) {
		tmp = Math.sin(re);
	} else {
		tmp = re + (re * (0.041666666666666664 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8e+30:
		tmp = math.sin(re)
	else:
		tmp = re + (re * (0.041666666666666664 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e+30)
		tmp = sin(re);
	else
		tmp = Float64(re + Float64(re * Float64(0.041666666666666664 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8e+30)
		tmp = sin(re);
	else
		tmp = re + (re * (0.041666666666666664 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e+30], N[Sin[re], $MachinePrecision], N[(re + N[(re * N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.8000000000000005e30

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.9%

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

   if 6.8000000000000005e30 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in im around 0 73.9%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 73.9%

         \[\leadsto re + \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re} + 0.5 \cdot \left({im}^{2} \cdot re\right)\right) \]

      2. associate-*r* 73.9%

         \[\leadsto re + \left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re}\right) \]

      3. distribute-rgt-out 73.9%

         \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right)} \]

      4. +-commutative 73.9%

         \[\leadsto re + re \cdot \color{blue}{\left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      5. unpow2 73.9%

         \[\leadsto re + re \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 0.041666666666666664 \cdot {im}^{4}\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   10. Taylor expanded in im around inf 73.9%

       \[\leadsto re + \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 73.9%

          \[\leadsto re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re} \]

       2. *-commutative 73.9%

          \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]

   12. Simplified 73.9%

       \[\leadsto re + \color{blue}{re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 8: 62.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 500.0)
   (sin re)
   (if (<= im 6e+183)
     (* (* re (* im im)) (+ 0.5 (* -0.08333333333333333 (* re re))))
     (* (* 0.5 re) (+ (* im im) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = sin(re);
	} else if (im <= 6e+183) {
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)));
	} else {
		tmp = (0.5 * re) * ((im * 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 <= 500.0d0) then
        tmp = sin(re)
    else if (im <= 6d+183) then
        tmp = (re * (im * im)) * (0.5d0 + ((-0.08333333333333333d0) * (re * re)))
    else
        tmp = (0.5d0 * re) * ((im * im) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = Math.sin(re);
	} else if (im <= 6e+183) {
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)));
	} else {
		tmp = (0.5 * re) * ((im * im) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 500.0:
		tmp = math.sin(re)
	elif im <= 6e+183:
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)))
	else:
		tmp = (0.5 * re) * ((im * im) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 500.0)
		tmp = sin(re);
	elseif (im <= 6e+183)
		tmp = Float64(Float64(re * Float64(im * im)) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re))));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 500.0)
		tmp = sin(re);
	elseif (im <= 6e+183)
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)));
	else
		tmp = (0.5 * re) * ((im * im) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 500.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6e+183], N[(N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 500

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.7%

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

   if 500 < im < 5.99999999999999992e183

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 20.3%

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

   6. Simplified 20.3%

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

   7. Taylor expanded in re around 0 6.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right)} \]
   8. Step-by-step derivation

      1. metadata-eval 6.9%

         \[\leadsto re \cdot \left(\color{blue}{0.5 \cdot 2} + 0.5 \cdot {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      2. distribute-lft-in 6.9%

         \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(2 + {im}^{2}\right)\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      3. associate-*l* 6.9%

         \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 + {im}^{2}\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      4. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(2 + {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      5. associate-*r* 6.9%

         \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      6. sub-neg 6.9%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(-0.08333333333333333 \cdot {im}^{2} + \left(-0.16666666666666666\right)\right)} \]

      7. +-commutative 6.9%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(\left(-0.16666666666666666\right) + -0.08333333333333333 \cdot {im}^{2}\right)} \]

      8. metadata-eval 6.9%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \left(\color{blue}{-0.16666666666666666} + -0.08333333333333333 \cdot {im}^{2}\right) \]

      9. metadata-eval 6.9%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \left(\color{blue}{-0.08333333333333333 \cdot 2} + -0.08333333333333333 \cdot {im}^{2}\right) \]

      10. distribute-lft-in 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(2 + {im}^{2}\right)\right)} \]

      11. associate-*l* 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right) \cdot \left(2 + {im}^{2}\right)} \]

      12. *-commutative 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right)} \cdot \left(2 + {im}^{2}\right) \]

      13. associate-*r* 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot \left(2 + {im}^{2}\right)\right)} \]

      14. unpow3 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(2 + {im}^{2}\right)\right) \]

      15. unpow2 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(2 + {im}^{2}\right)\right) \]

      16. associate-*l* 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\right)} \]

      17. associate-*r* 6.9%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)} \]

   10. Taylor expanded in im around inf 39.3%

       \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 39.3%

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot re\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

       2. *-commutative 39.3%

          \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot im\right)\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

   12. Simplified 39.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot im\right)\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

   if 5.99999999999999992e183 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 94.4%

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

   6. Simplified 94.4%

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

   7. Taylor expanded in im around 0 94.4%

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

      1. unpow2 94.4%

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

   9. Simplified 94.4%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]

Alternative 9: 48.2% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 114 \lor \neg \left(im \leq 5 \cdot 10^{+183}\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 114.0) (not (<= im 5e+183)))
   (* (* 0.5 re) (+ (* im im) 2.0))
   (* (* re (* im im)) (+ 0.5 (* -0.08333333333333333 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 114.0) || !(im <= 5e+183)) {
		tmp = (0.5 * re) * ((im * im) + 2.0);
	} else {
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (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 <= 114.0d0) .or. (.not. (im <= 5d+183))) then
        tmp = (0.5d0 * re) * ((im * im) + 2.0d0)
    else
        tmp = (re * (im * im)) * (0.5d0 + ((-0.08333333333333333d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 114.0) || !(im <= 5e+183)) {
		tmp = (0.5 * re) * ((im * im) + 2.0);
	} else {
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 114.0) or not (im <= 5e+183):
		tmp = (0.5 * re) * ((im * im) + 2.0)
	else:
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 114.0) || !(im <= 5e+183))
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(Float64(re * Float64(im * im)) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 114.0) || ~((im <= 5e+183)))
		tmp = (0.5 * re) * ((im * im) + 2.0);
	else
		tmp = (re * (im * im)) * (0.5 + (-0.08333333333333333 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 114.0], N[Not[LessEqual[im, 5e+183]], $MachinePrecision]], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 114 or 5.00000000000000009e183 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 64.2%

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

   6. Simplified 64.2%

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

   7. Taylor expanded in im around 0 52.1%

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

      1. unpow2 52.1%

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

   9. Simplified 52.1%

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

   if 114 < im < 5.00000000000000009e183

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 20.0%

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

   6. Simplified 20.0%

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

   7. Taylor expanded in re around 0 6.8%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right)} \]
   8. Step-by-step derivation

      1. metadata-eval 6.8%

         \[\leadsto re \cdot \left(\color{blue}{0.5 \cdot 2} + 0.5 \cdot {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      2. distribute-lft-in 6.8%

         \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(2 + {im}^{2}\right)\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      3. associate-*l* 6.8%

         \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 + {im}^{2}\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      4. *-commutative 6.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(2 + {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      5. associate-*r* 6.8%

         \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      6. sub-neg 6.8%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(-0.08333333333333333 \cdot {im}^{2} + \left(-0.16666666666666666\right)\right)} \]

      7. +-commutative 6.8%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(\left(-0.16666666666666666\right) + -0.08333333333333333 \cdot {im}^{2}\right)} \]

      8. metadata-eval 6.8%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \left(\color{blue}{-0.16666666666666666} + -0.08333333333333333 \cdot {im}^{2}\right) \]

      9. metadata-eval 6.8%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \left(\color{blue}{-0.08333333333333333 \cdot 2} + -0.08333333333333333 \cdot {im}^{2}\right) \]

      10. distribute-lft-in 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(2 + {im}^{2}\right)\right)} \]

      11. associate-*l* 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right) \cdot \left(2 + {im}^{2}\right)} \]

      12. *-commutative 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right)} \cdot \left(2 + {im}^{2}\right) \]

      13. associate-*r* 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot \left(2 + {im}^{2}\right)\right)} \]

      14. unpow3 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(2 + {im}^{2}\right)\right) \]

      15. unpow2 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(2 + {im}^{2}\right)\right) \]

      16. associate-*l* 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\right)} \]

      17. associate-*r* 6.8%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} \]

   9. Simplified 38.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)} \]

   10. Taylor expanded in im around inf 38.4%

       \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 38.4%

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot re\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

       2. *-commutative 38.4%

          \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot im\right)\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

   12. Simplified 38.4%

       \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot im\right)\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 114 \lor \neg \left(im \leq 5 \cdot 10^{+183}\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 10: 41.4% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+129}:\\ \;\;\;\;\left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.55e+129)
   (* (+ 0.5 (* -0.08333333333333333 (* re re))) (* re 2.0))
   (* (* 0.5 re) (+ (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.55e+129) {
		tmp = (0.5 + (-0.08333333333333333 * (re * re))) * (re * 2.0);
	} else {
		tmp = (0.5 * re) * ((im * 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 <= 1.55d+129) then
        tmp = (0.5d0 + ((-0.08333333333333333d0) * (re * re))) * (re * 2.0d0)
    else
        tmp = (0.5d0 * re) * ((im * im) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.55e+129) {
		tmp = (0.5 + (-0.08333333333333333 * (re * re))) * (re * 2.0);
	} else {
		tmp = (0.5 * re) * ((im * im) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.55e+129:
		tmp = (0.5 + (-0.08333333333333333 * (re * re))) * (re * 2.0)
	else:
		tmp = (0.5 * re) * ((im * im) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.55e+129)
		tmp = Float64(Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re))) * Float64(re * 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.55e+129)
		tmp = (0.5 + (-0.08333333333333333 * (re * re))) * (re * 2.0);
	else
		tmp = (0.5 * re) * ((im * im) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.55e+129], N[(N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.55e129

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in re around 0 33.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right)} \]
   8. Step-by-step derivation

      1. metadata-eval 33.7%

         \[\leadsto re \cdot \left(\color{blue}{0.5 \cdot 2} + 0.5 \cdot {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      2. distribute-lft-in 33.7%

         \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(2 + {im}^{2}\right)\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      3. associate-*l* 33.7%

         \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 + {im}^{2}\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      4. *-commutative 33.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(2 + {im}^{2}\right) + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      5. associate-*r* 33.6%

         \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} + {re}^{3} \cdot \left(-0.08333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) \]

      6. sub-neg 33.6%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(-0.08333333333333333 \cdot {im}^{2} + \left(-0.16666666666666666\right)\right)} \]

      7. +-commutative 33.6%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(\left(-0.16666666666666666\right) + -0.08333333333333333 \cdot {im}^{2}\right)} \]

      8. metadata-eval 33.6%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \left(\color{blue}{-0.16666666666666666} + -0.08333333333333333 \cdot {im}^{2}\right) \]

      9. metadata-eval 33.6%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \left(\color{blue}{-0.08333333333333333 \cdot 2} + -0.08333333333333333 \cdot {im}^{2}\right) \]

      10. distribute-lft-in 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + {re}^{3} \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(2 + {im}^{2}\right)\right)} \]

      11. associate-*l* 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right) \cdot \left(2 + {im}^{2}\right)} \]

      12. *-commutative 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right)} \cdot \left(2 + {im}^{2}\right) \]

      13. associate-*r* 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot \left(2 + {im}^{2}\right)\right)} \]

      14. unpow3 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(2 + {im}^{2}\right)\right) \]

      15. unpow2 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(2 + {im}^{2}\right)\right) \]

      16. associate-*l* 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + -0.08333333333333333 \cdot \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\right)} \]

      17. associate-*r* 33.6%

          \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} \]

   9. Simplified 49.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)} \]

   10. Taylor expanded in im around 0 39.1%

       \[\leadsto \color{blue}{\left(2 \cdot re\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative 39.1%

          \[\leadsto \color{blue}{\left(re \cdot 2\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

   12. Simplified 39.1%

       \[\leadsto \color{blue}{\left(re \cdot 2\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \]

   if 1.55e129 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in im around 0 78.7%

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

      1. unpow2 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+129}:\\ \;\;\;\;\left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right) \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]

Alternative 11: 28.8% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.5e+43) re (+ 0.08333333333333333 (/ 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.5e+43) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + (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 <= 4.5d+43) then
        tmp = re
    else
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.5e+43) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.5e+43:
		tmp = re
	else:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.5e+43)
		tmp = re;
	else
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.5e+43)
		tmp = re;
	else
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.5e+43], re, N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.5e43

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in im around 0 33.2%

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

   if 4.5e43 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 6.4%

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

   6. Taylor expanded in re around 0 6.4%

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

      1. associate-*r/ 6.4%

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

      2. metadata-eval 6.4%

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

      3. unpow2 6.4%

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

   8. Simplified 6.4%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 12: 47.3% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) (+ (* im im) 2.0)))

C

double code(double re, double im) {
	return (0.5 * re) * ((im * im) + 2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * ((im * im) + 2.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * ((im * im) + 2.0);
}

Python

def code(re, im):
	return (0.5 * re) * ((im * im) + 2.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(Float64(im * im) + 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * ((im * im) + 2.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $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-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in im around 0 48.6%

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

   1. unpow2 48.6%

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

9. Simplified 48.6%

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

10. Final simplification 48.6%

    \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot im + 2\right) \]

Alternative 13: 28.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 2.2e+44) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2e+44) {
		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 <= 2.2d+44) 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 <= 2.2e+44) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2e+44:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2e+44)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2e+44)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2e+44], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.19999999999999996e44

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in im around 0 33.2%

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

   if 2.19999999999999996e44 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 6.4%

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

   6. Taylor expanded in re around 0 6.2%

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

      1. unpow2 6.2%

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

   8. Simplified 6.2%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 14: 4.2% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

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

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 6.4%

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

6. Taylor expanded in re around 0 6.4%

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

   1. associate-*r/ 6.4%

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

   2. metadata-eval 6.4%

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

   3. unpow2 6.4%

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

8. Simplified 6.4%

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

9. Taylor expanded in re around inf 4.2%

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

10. Final simplification 4.2%

    \[\leadsto 0.08333333333333333 \]

Alternative 15: 26.4% 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-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in im around 0 27.9%

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

8. Final simplification 27.9%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023275 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))