math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

Alternatives: 16

Speedup: 1.0×

• 48.2% 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: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 17:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \sqrt{{im}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 17.0)
   (+ (sin re) (* 0.5 (* (sin re) (* im im))))
   (if (<= im 1.2e+77)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (* 0.5 (sin re)) (sqrt (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 17.0) {
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	} else if (im <= 1.2e+77) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * sqrt(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 <= 17.0d0) then
        tmp = sin(re) + (0.5d0 * (sin(re) * (im * im)))
    else if (im <= 1.2d+77) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * sqrt((im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 17.0) {
		tmp = Math.sin(re) + (0.5 * (Math.sin(re) * (im * im)));
	} else if (im <= 1.2e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * Math.sqrt(Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 17.0:
		tmp = math.sin(re) + (0.5 * (math.sin(re) * (im * im)))
	elif im <= 1.2e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * math.sqrt(math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 17.0)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(sin(re) * Float64(im * im))));
	elseif (im <= 1.2e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * sqrt((im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 17.0)
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	elseif (im <= 1.2e+77)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * sqrt((im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 17.0], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], 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[Sqrt[N[Power[im, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 17

   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 82.4%

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

      1. unpow2 49.3%

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

   6. Applied egg-rr 82.4%

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

   if 17 < im < 1.1999999999999999e77

   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 80.0%

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

   6. Simplified 80.0%

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

   if 1.1999999999999999e77 < 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 75.9%

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

   5. Taylor expanded in im around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-*r* 75.9%

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

      3. *-commutative 75.9%

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

   7. Simplified 75.9%

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

      1. add-sqr-sqrt 75.9%

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

      2. sqrt-unprod 100.0%

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

      3. pow-prod-up 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 17:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \sqrt{{im}^{4}}\\ \end{array} \]

Alternative 3: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 17:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 17.0)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.45e+151)
       (* (+ (exp (- im)) (exp im)) (* 0.5 re))
       (* t_0 (* im im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 17.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.45e+151) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 17.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.45e+151)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 17.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.45e+151], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 17

   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 82.4%

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

   6. Simplified 82.4%

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

   if 17 < im < 1.45000000000000009e151

   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 73.9%

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

   6. Simplified 73.9%

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

   if 1.45000000000000009e151 < 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 97.9%

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

   5. Taylor expanded in im around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*r* 97.9%

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

      3. *-commutative 97.9%

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

   7. Simplified 97.9%

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

      1. unpow2 71.4%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 17:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 4: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 17:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 17.0)
   (+ (sin re) (* 0.5 (* (sin re) (* im im))))
   (if (<= im 1.45e+151)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (* 0.5 (sin re)) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 17.0) {
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	} else if (im <= 1.45e+151) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 17.0d0) then
        tmp = sin(re) + (0.5d0 * (sin(re) * (im * im)))
    else if (im <= 1.45d+151) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 17.0) {
		tmp = Math.sin(re) + (0.5 * (Math.sin(re) * (im * im)));
	} else if (im <= 1.45e+151) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 17.0:
		tmp = math.sin(re) + (0.5 * (math.sin(re) * (im * im)))
	elif im <= 1.45e+151:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 17.0)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(sin(re) * Float64(im * im))));
	elseif (im <= 1.45e+151)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 17.0)
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	elseif (im <= 1.45e+151)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 17.0], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.45e+151], 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[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 17

   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 82.4%

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

      1. unpow2 49.3%

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

   6. Applied egg-rr 82.4%

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

   if 17 < im < 1.45000000000000009e151

   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 73.9%

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

   6. Simplified 73.9%

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

   if 1.45000000000000009e151 < 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 97.9%

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

   5. Taylor expanded in im around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*r* 97.9%

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

      3. *-commutative 97.9%

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

   7. Simplified 97.9%

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

      1. unpow2 71.4%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 17:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 5: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 4500000:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 4500000.0)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.4e+154)
       (* (fma im im 2.0) (+ (* 0.5 re) (* re (* re -0.16666666666666666))))
       (* t_0 (* im im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 4500000.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.4e+154) {
		tmp = fma(im, im, 2.0) * ((0.5 * re) + (re * (re * -0.16666666666666666)));
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 4500000.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.4e+154)
		tmp = Float64(fma(im, im, 2.0) * Float64(Float64(0.5 * re) + Float64(re * Float64(re * -0.16666666666666666))));
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4500000.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(0.5 * re), $MachinePrecision] + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5e6

   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 82.0%

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

   6. Simplified 82.0%

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

   if 4.5e6 < im < 1.4e154

   1. Initial program 100.0%

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

      1. distribute-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 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in re around 0 6.8%

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

      1. +-commutative 6.8%

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

      2. associate-*r* 6.8%

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

      3. associate-*r* 6.8%

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

      4. distribute-rgt-out 24.2%

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

      5. +-commutative 24.2%

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

      6. unpow2 24.2%

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

      7. fma-udef 24.2%

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

      8. *-commutative 24.2%

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

      9. *-commutative 24.2%

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

   9. Simplified 24.2%

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

   11. Applied egg-rr 0.0%

       \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{e^{\log re + \log \left(re \cdot -0.16666666666666666\right)}}\right) \]
   12. Step-by-step derivation

       1. exp-sum 0.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{e^{\log re} \cdot e^{\log \left(re \cdot -0.16666666666666666\right)}}\right) \]

       2. rem-exp-log 13.9%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{re} \cdot e^{\log \left(re \cdot -0.16666666666666666\right)}\right) \]

       3. rem-exp-log 28.3%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]

   13. Simplified 28.3%

       \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{re \cdot \left(re \cdot -0.16666666666666666\right)}\right) \]

   if 1.4e154 < im

   1. Initial program 100.0%

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

      1. distribute-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%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. unpow2 73.2%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4500000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 6: 66.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 400.0)
   (sin re)
   (if (<= im 1.4e+154)
     (* (fma im im 2.0) (+ (* 0.5 re) (* re (* re -0.16666666666666666))))
     (* (* 0.5 (sin re)) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 400.0) {
		tmp = sin(re);
	} else if (im <= 1.4e+154) {
		tmp = fma(im, im, 2.0) * ((0.5 * re) + (re * (re * -0.16666666666666666)));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 400.0)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = Float64(fma(im, im, 2.0) * Float64(Float64(0.5 * re) + Float64(re * Float64(re * -0.16666666666666666))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 400.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(0.5 * re), $MachinePrecision] + N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 400

   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 66.2%

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

   if 400 < im < 1.4e154

   1. Initial program 100.0%

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

      1. distribute-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 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 6.7%

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

      1. +-commutative 6.7%

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

      2. associate-*r* 6.7%

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

      3. associate-*r* 6.7%

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

      4. distribute-rgt-out 23.3%

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

      5. +-commutative 23.3%

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

      6. unpow2 23.3%

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

      7. fma-udef 23.3%

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

      8. *-commutative 23.3%

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

      9. *-commutative 23.3%

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

   9. Simplified 23.3%

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

   11. Applied egg-rr 0.0%

       \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{e^{\log re + \log \left(re \cdot -0.16666666666666666\right)}}\right) \]
   12. Step-by-step derivation

       1. exp-sum 0.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{e^{\log re} \cdot e^{\log \left(re \cdot -0.16666666666666666\right)}}\right) \]

       2. rem-exp-log 13.4%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{re} \cdot e^{\log \left(re \cdot -0.16666666666666666\right)}\right) \]

       3. rem-exp-log 27.2%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}\right) \]

   13. Simplified 27.2%

       \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5 + \color{blue}{re \cdot \left(re \cdot -0.16666666666666666\right)}\right) \]

   if 1.4e154 < im

   1. Initial program 100.0%

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

      1. distribute-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%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. unpow2 73.2%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re + re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 210000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 210000.0)
   (sin re)
   (if (<= im 1.4e+154)
     (+ re (* (* re -0.16666666666666666) (* re 0.16666666666666666)))
     (* (* 0.5 (sin re)) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 210000.0) {
		tmp = sin(re);
	} else if (im <= 1.4e+154) {
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 210000.0d0) then
        tmp = sin(re)
    else if (im <= 1.4d+154) then
        tmp = re + ((re * (-0.16666666666666666d0)) * (re * 0.16666666666666666d0))
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 210000.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.4e+154) {
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 210000.0:
		tmp = math.sin(re)
	elif im <= 1.4e+154:
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666))
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 210000.0)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = Float64(re + Float64(Float64(re * -0.16666666666666666) * Float64(re * 0.16666666666666666)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 210000.0)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 210000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re + N[(N[(re * -0.16666666666666666), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.1e5

   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 66.2%

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

   if 2.1e5 < im < 1.4e154

   1. Initial program 100.0%

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

      1. distribute-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 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 6.7%

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

      1. +-commutative 6.7%

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

      2. associate-*r* 6.7%

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

      3. associate-*r* 6.7%

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

      4. distribute-rgt-out 23.3%

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

      5. +-commutative 23.3%

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

      6. unpow2 23.3%

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

      7. fma-udef 23.3%

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

      8. *-commutative 23.3%

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

      9. *-commutative 23.3%

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

   9. Simplified 23.3%

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

   11. Applied egg-rr 22.6%

       \[\leadsto \color{blue}{re + \left(-re \cdot -0.16666666666666666\right) \cdot \left(re \cdot -0.16666666666666666\right)} \]
   12. Step-by-step derivation

       1. *-commutative 22.6%

          \[\leadsto re + \color{blue}{\left(re \cdot -0.16666666666666666\right) \cdot \left(-re \cdot -0.16666666666666666\right)} \]

       2. distribute-rgt-neg-in 22.6%

          \[\leadsto re + \left(re \cdot -0.16666666666666666\right) \cdot \color{blue}{\left(re \cdot \left(--0.16666666666666666\right)\right)} \]

       3. metadata-eval 22.6%

          \[\leadsto re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot \color{blue}{0.16666666666666666}\right) \]

   13. Simplified 22.6%

       \[\leadsto \color{blue}{re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)} \]

   if 1.4e154 < im

   1. Initial program 100.0%

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

      1. distribute-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%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. unpow2 73.2%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 210000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 65.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;re + -0.16666666666666666 \cdot {re}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2e+14)
   (sin re)
   (if (<= im 1.4e+154)
     (+ re (* -0.16666666666666666 (pow re 3.0)))
     (* (* 0.5 (sin re)) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2e+14) {
		tmp = sin(re);
	} else if (im <= 1.4e+154) {
		tmp = re + (-0.16666666666666666 * pow(re, 3.0));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.2d+14) then
        tmp = sin(re)
    else if (im <= 1.4d+154) then
        tmp = re + ((-0.16666666666666666d0) * (re ** 3.0d0))
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.2e+14) {
		tmp = Math.sin(re);
	} else if (im <= 1.4e+154) {
		tmp = re + (-0.16666666666666666 * Math.pow(re, 3.0));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.2e+14:
		tmp = math.sin(re)
	elif im <= 1.4e+154:
		tmp = re + (-0.16666666666666666 * math.pow(re, 3.0))
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2e+14)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.2e+14)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = re + (-0.16666666666666666 * (re ^ 3.0));
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2e+14], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.2e14

   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 65.8%

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

   if 4.2e14 < im < 1.4e154

   1. Initial program 100.0%

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

      1. distribute-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 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in re around 0 6.8%

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

      1. +-commutative 6.8%

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

      2. associate-*r* 6.8%

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

      3. associate-*r* 6.8%

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

      4. distribute-rgt-out 24.2%

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

      5. +-commutative 24.2%

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

      6. unpow2 24.2%

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

      7. fma-udef 24.2%

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

      8. *-commutative 24.2%

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

      9. *-commutative 24.2%

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

   9. Simplified 24.2%

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

   10. Taylor expanded in im around 0 23.4%

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

       1. distribute-rgt-in 23.4%

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

       2. *-commutative 23.4%

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

       3. associate-*l* 23.4%

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

       4. metadata-eval 23.4%

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

       5. *-commutative 23.4%

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

       6. associate-*l* 23.4%

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

       7. metadata-eval 23.4%

          \[\leadsto {re}^{3} \cdot -0.16666666666666666 + re \cdot \color{blue}{1} \]

   12. Simplified 23.4%

       \[\leadsto \color{blue}{{re}^{3} \cdot -0.16666666666666666 + re \cdot 1} \]

   if 1.4e154 < im

   1. Initial program 100.0%

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

      1. distribute-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%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. unpow2 73.2%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;re + -0.16666666666666666 \cdot {re}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 9: 62.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9500.0)
   (sin re)
   (if (<= im 1.4e+154)
     (+ re (* (* re -0.16666666666666666) (* re 0.16666666666666666)))
     (+ re (* 0.5 (* re (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9500.0) {
		tmp = sin(re);
	} else if (im <= 1.4e+154) {
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	} else {
		tmp = re + (0.5 * (re * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9500.0d0) then
        tmp = sin(re)
    else if (im <= 1.4d+154) then
        tmp = re + ((re * (-0.16666666666666666d0)) * (re * 0.16666666666666666d0))
    else
        tmp = re + (0.5d0 * (re * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9500.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.4e+154) {
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	} else {
		tmp = re + (0.5 * (re * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9500.0:
		tmp = math.sin(re)
	elif im <= 1.4e+154:
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666))
	else:
		tmp = re + (0.5 * (re * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9500.0)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = Float64(re + Float64(Float64(re * -0.16666666666666666) * Float64(re * 0.16666666666666666)));
	else
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9500.0)
		tmp = sin(re);
	elseif (im <= 1.4e+154)
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	else
		tmp = re + (0.5 * (re * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9500.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re + N[(N[(re * -0.16666666666666666), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9500

   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 66.2%

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

   if 9500 < im < 1.4e154

   1. Initial program 100.0%

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

      1. distribute-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 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 6.7%

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

      1. +-commutative 6.7%

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

      2. associate-*r* 6.7%

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

      3. associate-*r* 6.7%

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

      4. distribute-rgt-out 23.3%

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

      5. +-commutative 23.3%

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

      6. unpow2 23.3%

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

      7. fma-udef 23.3%

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

      8. *-commutative 23.3%

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

      9. *-commutative 23.3%

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

   9. Simplified 23.3%

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

   11. Applied egg-rr 22.6%

       \[\leadsto \color{blue}{re + \left(-re \cdot -0.16666666666666666\right) \cdot \left(re \cdot -0.16666666666666666\right)} \]
   12. Step-by-step derivation

       1. *-commutative 22.6%

          \[\leadsto re + \color{blue}{\left(re \cdot -0.16666666666666666\right) \cdot \left(-re \cdot -0.16666666666666666\right)} \]

       2. distribute-rgt-neg-in 22.6%

          \[\leadsto re + \left(re \cdot -0.16666666666666666\right) \cdot \color{blue}{\left(re \cdot \left(--0.16666666666666666\right)\right)} \]

       3. metadata-eval 22.6%

          \[\leadsto re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot \color{blue}{0.16666666666666666}\right) \]

   13. Simplified 22.6%

       \[\leadsto \color{blue}{re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)} \]

   if 1.4e154 < im

   1. Initial program 100.0%

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

      1. distribute-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%

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

   7. Taylor expanded in re around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-*l* 73.2%

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

      4. +-commutative 73.2%

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

      5. unpow2 73.2%

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

      6. fma-udef 73.2%

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

   9. Simplified 73.2%

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

   10. Taylor expanded in im around 0 73.2%

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

       1. unpow2 73.2%

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

   12. Applied egg-rr 73.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 48.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.8 \cdot 10^{+133}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.8e+133)
   (+ re (* 0.5 (* re (* im im))))
   (+ re (* (* re -0.16666666666666666) (* re 0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.8e+133) {
		tmp = re + (0.5 * (re * (im * im)));
	} else {
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8.8d+133) then
        tmp = re + (0.5d0 * (re * (im * im)))
    else
        tmp = re + ((re * (-0.16666666666666666d0)) * (re * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8.8e+133) {
		tmp = re + (0.5 * (re * (im * im)));
	} else {
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.8e+133:
		tmp = re + (0.5 * (re * (im * im)))
	else:
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.8e+133)
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	else
		tmp = Float64(re + Float64(Float64(re * -0.16666666666666666) * Float64(re * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8.8e+133)
		tmp = re + (0.5 * (re * (im * im)));
	else
		tmp = re + ((re * -0.16666666666666666) * (re * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.8e+133], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(N[(re * -0.16666666666666666), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.8e133

   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 78.3%

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

   6. Simplified 78.3%

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

   7. Taylor expanded in re around 0 56.2%

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

      1. *-commutative 56.2%

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

      2. *-commutative 56.2%

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

      3. associate-*l* 56.2%

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

      4. +-commutative 56.2%

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

      5. unpow2 56.2%

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

      6. fma-udef 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in im around 0 56.2%

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

       1. unpow2 56.2%

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

   12. Applied egg-rr 56.2%

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

   if 8.8e133 < re

   1. Initial program 100.0%

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

      1. distribute-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 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in re around 0 5.2%

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

      1. +-commutative 5.2%

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

      2. associate-*r* 5.2%

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

      3. associate-*r* 5.2%

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

      4. distribute-rgt-out 37.8%

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

      5. +-commutative 37.8%

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

      6. unpow2 37.8%

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

      7. fma-udef 37.8%

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

      8. *-commutative 37.8%

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

      9. *-commutative 37.8%

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

   9. Simplified 37.8%

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

   11. Applied egg-rr 35.8%

       \[\leadsto \color{blue}{re + \left(-re \cdot -0.16666666666666666\right) \cdot \left(re \cdot -0.16666666666666666\right)} \]
   12. Step-by-step derivation

       1. *-commutative 35.8%

          \[\leadsto re + \color{blue}{\left(re \cdot -0.16666666666666666\right) \cdot \left(-re \cdot -0.16666666666666666\right)} \]

       2. distribute-rgt-neg-in 35.8%

          \[\leadsto re + \left(re \cdot -0.16666666666666666\right) \cdot \color{blue}{\left(re \cdot \left(--0.16666666666666666\right)\right)} \]

       3. metadata-eval 35.8%

          \[\leadsto re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot \color{blue}{0.16666666666666666}\right) \]

   13. Simplified 35.8%

       \[\leadsto \color{blue}{re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.8 \cdot 10^{+133}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(re \cdot -0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 11: 30.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 20000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 20000000.0) re (* (/ 0.5 re) (/ 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 20000000.0) {
		tmp = re;
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 20000000.0d0) then
        tmp = re
    else
        tmp = (0.5d0 / re) * (0.5d0 / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 20000000.0) {
		tmp = re;
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 20000000.0:
		tmp = re
	else:
		tmp = (0.5 / re) * (0.5 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 20000000.0)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 / re) * Float64(0.5 / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 20000000.0)
		tmp = re;
	else
		tmp = (0.5 / re) * (0.5 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 20000000.0], re, N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2e7

   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 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in re around 0 49.0%

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

      1. *-commutative 49.0%

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

      2. *-commutative 49.0%

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

      3. associate-*l* 49.0%

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

      4. +-commutative 49.0%

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

      5. unpow2 49.0%

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

      6. fma-udef 49.0%

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

   9. Simplified 49.0%

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

   10. Taylor expanded in im around 0 36.1%

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

   if 2e7 < 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 17.0%

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

   6. Taylor expanded in re around 0 17.0%

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

      1. add-sqr-sqrt 17.0%

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

      2. sqrt-div 17.0%

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

      3. metadata-eval 17.0%

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

      4. unpow2 17.0%

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

      5. sqrt-prod 16.6%

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

      6. add-sqr-sqrt 30.2%

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

      7. sqrt-div 30.2%

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

      8. metadata-eval 30.2%

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

      9. unpow2 30.2%

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

      10. sqrt-prod 16.6%

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

      11. add-sqr-sqrt 17.0%

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

   8. Applied egg-rr 17.0%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 20000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]

Alternative 12: 48.8% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $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. Taylor expanded in im around 0 77.9%

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

6. Simplified 77.9%

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

7. Taylor expanded in re around 0 49.5%

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

   1. *-commutative 49.5%

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

   2. *-commutative 49.5%

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

   3. associate-*l* 49.5%

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

   4. +-commutative 49.5%

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

   5. unpow2 49.5%

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

   6. fma-udef 49.5%

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

9. Simplified 49.5%

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

10. Taylor expanded in im around 0 49.5%

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

    1. unpow2 49.5%

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

12. Applied egg-rr 49.5%

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

13. Final simplification 49.5%

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

Alternative 13: 30.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+27}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re + -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2e+27) re (* re (+ re -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+27) {
		tmp = re;
	} else {
		tmp = re * (re + -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.2d+27) then
        tmp = re
    else
        tmp = re * (re + (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+27) {
		tmp = re;
	} else {
		tmp = re * (re + -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.2e+27:
		tmp = re
	else:
		tmp = re * (re + -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2e+27)
		tmp = re;
	else
		tmp = Float64(re * Float64(re + -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.2e+27)
		tmp = re;
	else
		tmp = re * (re + -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2e+27], re, N[(re * N[(re + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.20000000000000018e27

   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 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in re around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. *-commutative 48.5%

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

      3. associate-*l* 48.5%

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

      4. +-commutative 48.5%

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

      5. unpow2 48.5%

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

      6. fma-udef 48.5%

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

   9. Simplified 48.5%

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

   10. Taylor expanded in im around 0 35.7%

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

   if 5.20000000000000018e27 < 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 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in re around 0 2.4%

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

      1. +-commutative 2.4%

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

      2. associate-*r* 2.4%

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

      3. associate-*r* 2.4%

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

      4. distribute-rgt-out 60.5%

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

      5. +-commutative 60.5%

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

      6. unpow2 60.5%

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

      7. fma-udef 60.5%

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

      8. *-commutative 60.5%

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

      9. *-commutative 60.5%

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

   9. Simplified 60.5%

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

   11. Applied egg-rr 9.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, re \cdot -0.16666666666666666\right)} \]
   12. Step-by-step derivation

       1. fma-udef 9.3%

          \[\leadsto \color{blue}{re \cdot re + re \cdot -0.16666666666666666} \]

       2. distribute-lft-out 9.3%

          \[\leadsto \color{blue}{re \cdot \left(re + -0.16666666666666666\right)} \]

   13. Simplified 9.3%

       \[\leadsto \color{blue}{re \cdot \left(re + -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+27}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re + -0.16666666666666666\right)\\ \end{array} \]

Alternative 14: 4.3% 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 11.7%

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

6. Taylor expanded in re around 0 11.5%

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

   1. associate-*r/ 11.5%

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

   2. metadata-eval 11.5%

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

8. Simplified 11.5%

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

9. Taylor expanded in re around inf 4.4%

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

10. Final simplification 4.4%

    \[\leadsto 0.08333333333333333 \]

Alternative 15: 4.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. distribute-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 61.9%

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

6. Simplified 61.9%

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

8. Applied egg-rr 4.8%

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

   1. *-inverses 4.8%

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

10. Simplified 4.8%

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

11. Final simplification 4.8%

    \[\leadsto 1 \]

Alternative 16: 27.5% 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 im around 0 77.9%

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

6. Simplified 77.9%

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

7. Taylor expanded in re around 0 49.5%

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

   1. *-commutative 49.5%

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

   2. *-commutative 49.5%

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

   3. associate-*l* 49.5%

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

   4. +-commutative 49.5%

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

   5. unpow2 49.5%

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

   6. fma-udef 49.5%

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

9. Simplified 49.5%

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

10. Taylor expanded in im around 0 27.7%

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

11. Final simplification 27.7%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023311 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))