math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 14

Speedup: 1.0×

• 49.5% 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(im) + exp(Float64(-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. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 68.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \sqrt{{im}^{4} \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.1e+19)
   (sin re)
   (if (<= im 1.15e+77)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (if (<= im 1.35e+154)
       (* re (sqrt (* (pow im 4.0) 0.25)))
       (* 0.5 (* (sin re) (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.1e+19) {
		tmp = sin(re);
	} else if (im <= 1.15e+77) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else if (im <= 1.35e+154) {
		tmp = re * sqrt((pow(im, 4.0) * 0.25));
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.1d+19) then
        tmp = sin(re)
    else if (im <= 1.15d+77) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else if (im <= 1.35d+154) then
        tmp = re * sqrt(((im ** 4.0d0) * 0.25d0))
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.1e+19) {
		tmp = Math.sin(re);
	} else if (im <= 1.15e+77) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else if (im <= 1.35e+154) {
		tmp = re * Math.sqrt((Math.pow(im, 4.0) * 0.25));
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.1e+19:
		tmp = math.sin(re)
	elif im <= 1.15e+77:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	elif im <= 1.35e+154:
		tmp = re * math.sqrt((math.pow(im, 4.0) * 0.25))
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.1e+19)
		tmp = sin(re);
	elseif (im <= 1.15e+77)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	elseif (im <= 1.35e+154)
		tmp = Float64(re * sqrt(Float64((im ^ 4.0) * 0.25)));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.1e+19)
		tmp = sin(re);
	elseif (im <= 1.15e+77)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	elseif (im <= 1.35e+154)
		tmp = re * sqrt(((im ^ 4.0) * 0.25));
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.1e+19], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[Sqrt[N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 3.1e19

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.3%

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

   if 3.1e19 < im < 1.14999999999999997e77

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. +-commutative 3.3%

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

      2. unpow2 3.3%

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

      3. fma-define 3.3%

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

   7. Simplified 3.3%

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

   8. Taylor expanded in re around 0 17.7%

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

      1. associate-*r* 17.7%

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

      2. associate-*r* 17.7%

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

      3. *-commutative 17.7%

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

      4. distribute-rgt-out 25.4%

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

      5. +-commutative 25.4%

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

      6. unpow2 25.4%

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

      7. fma-undefine 25.4%

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

      8. *-commutative 25.4%

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

   10. Simplified 25.4%

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

   11. Taylor expanded in im around 0 24.9%

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

       1. +-commutative 24.9%

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

       2. *-commutative 24.9%

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

       3. distribute-rgt-in 24.9%

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

       4. associate-*l* 24.9%

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

       5. metadata-eval 24.9%

          \[\leadsto re \cdot \color{blue}{1} + \left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot 2 \]

       6. *-rgt-identity 24.9%

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

       7. *-commutative 24.9%

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

       8. associate-*l* 24.9%

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

       9. metadata-eval 24.9%

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

   13. Simplified 24.9%

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

   if 1.14999999999999997e77 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 73.7%

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

      1. associate-*r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in im around 0 13.0%

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

   9. Taylor expanded in im around inf 13.0%

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

       1. associate-*r* 13.0%

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

       2. *-commutative 13.0%

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

   11. Simplified 13.0%

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

       1. add-sqr-sqrt 13.0%

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

       2. sqrt-unprod 73.7%

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

       3. *-commutative 73.7%

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

       4. *-commutative 73.7%

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

       5. swap-sqr 73.7%

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

       6. pow-prod-up 73.7%

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

       7. metadata-eval 73.7%

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

       8. metadata-eval 73.7%

          \[\leadsto re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 73.7%

       \[\leadsto re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \sqrt{{im}^{4} \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \sqrt{{im}^{4} \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2e+19)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 1.16e+77)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (if (<= im 5e+152)
       (* re (sqrt (* (pow im 4.0) 0.25)))
       (* 0.5 (* (sin re) (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+19) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 1.16e+77) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else if (im <= 5e+152) {
		tmp = re * sqrt((pow(im, 4.0) * 0.25));
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2e+19)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 1.16e+77)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	elseif (im <= 5e+152)
		tmp = Float64(re * sqrt(Float64((im ^ 4.0) * 0.25)));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2e+19], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.16e+77], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+152], N[(re * N[Sqrt[N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 5.2e19

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. fma-define 81.7%

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

   7. Simplified 81.7%

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

   if 5.2e19 < im < 1.1600000000000001e77

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. +-commutative 3.3%

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

      2. unpow2 3.3%

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

      3. fma-define 3.3%

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

   7. Simplified 3.3%

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

   8. Taylor expanded in re around 0 17.7%

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

      1. associate-*r* 17.7%

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

      2. associate-*r* 17.7%

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

      3. *-commutative 17.7%

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

      4. distribute-rgt-out 25.4%

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

      5. +-commutative 25.4%

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

      6. unpow2 25.4%

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

      7. fma-undefine 25.4%

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

      8. *-commutative 25.4%

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

   10. Simplified 25.4%

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

   11. Taylor expanded in im around 0 24.9%

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

       1. +-commutative 24.9%

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

       2. *-commutative 24.9%

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

       3. distribute-rgt-in 24.9%

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

       4. associate-*l* 24.9%

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

       5. metadata-eval 24.9%

          \[\leadsto re \cdot \color{blue}{1} + \left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot 2 \]

       6. *-rgt-identity 24.9%

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

       7. *-commutative 24.9%

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

       8. associate-*l* 24.9%

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

       9. metadata-eval 24.9%

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

   13. Simplified 24.9%

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

   if 1.1600000000000001e77 < im < 5e152

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 73.7%

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

      1. associate-*r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in im around 0 13.0%

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

   9. Taylor expanded in im around inf 13.0%

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

       1. associate-*r* 13.0%

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

       2. *-commutative 13.0%

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

   11. Simplified 13.0%

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

       1. add-sqr-sqrt 13.0%

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

       2. sqrt-unprod 73.7%

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

       3. *-commutative 73.7%

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

       4. *-commutative 73.7%

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

       5. swap-sqr 73.7%

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

       6. pow-prod-up 73.7%

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

       7. metadata-eval 73.7%

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

       8. metadata-eval 73.7%

          \[\leadsto re \cdot \sqrt{{im}^{4} \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 73.7%

       \[\leadsto re \cdot \color{blue}{\sqrt{{im}^{4} \cdot 0.25}} \]

   if 5e152 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \sqrt{{im}^{4} \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;12 + \sin re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.6e-8)
   (sin re)
   (if (<= im 2e+51)
     (* (+ (exp im) (exp (- im))) (* 0.5 re))
     (+ 12.0 (* (sin re) (+ (* 0.001388888888888889 (pow im 6.0)) 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-8) {
		tmp = sin(re);
	} else if (im <= 2e+51) {
		tmp = (exp(im) + exp(-im)) * (0.5 * re);
	} else {
		tmp = 12.0 + (sin(re) * ((0.001388888888888889 * pow(im, 6.0)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.6d-8) then
        tmp = sin(re)
    else if (im <= 2d+51) then
        tmp = (exp(im) + exp(-im)) * (0.5d0 * re)
    else
        tmp = 12.0d0 + (sin(re) * ((0.001388888888888889d0 * (im ** 6.0d0)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-8) {
		tmp = Math.sin(re);
	} else if (im <= 2e+51) {
		tmp = (Math.exp(im) + Math.exp(-im)) * (0.5 * re);
	} else {
		tmp = 12.0 + (Math.sin(re) * ((0.001388888888888889 * Math.pow(im, 6.0)) + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.6e-8:
		tmp = math.sin(re)
	elif im <= 2e+51:
		tmp = (math.exp(im) + math.exp(-im)) * (0.5 * re)
	else:
		tmp = 12.0 + (math.sin(re) * ((0.001388888888888889 * math.pow(im, 6.0)) + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.6e-8)
		tmp = sin(re);
	elseif (im <= 2e+51)
		tmp = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(12.0 + Float64(sin(re) * Float64(Float64(0.001388888888888889 * (im ^ 6.0)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.6e-8)
		tmp = sin(re);
	elseif (im <= 2e+51)
		tmp = (exp(im) + exp(-im)) * (0.5 * re);
	else
		tmp = 12.0 + (sin(re) * ((0.001388888888888889 * (im ^ 6.0)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.6e-8], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2e+51], N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(12.0 + N[(N[Sin[re], $MachinePrecision] * N[(N[(0.001388888888888889 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.5999999999999999e-8

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.7%

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

   if 5.5999999999999999e-8 < im < 2e51

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. neg-sub0 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in re around 0 78.2%

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

      1. associate-*r* 78.2%

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

   7. Simplified 78.2%

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

   if 2e51 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + \left(0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\sin re + 0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right)\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left(\sin re + 0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right)\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re} + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left(\sin re + 0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right)\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re}\right) + \left(\sin re + 0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right)\right) \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right)} + \left(\sin re + 0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right)\right) \]

      6. associate-*r* 100.0%

         \[\leadsto \sin re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right) + \left(\sin re + \color{blue}{\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \sin re}\right) \]

      7. distribute-rgt1-in 100.0%

         \[\leadsto \sin re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right) + \color{blue}{\left(0.001388888888888889 \cdot {im}^{6} + 1\right) \cdot \sin re} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.041666666666666664 \cdot {im}^{4} + 0.5 \cdot {im}^{2}\right) + \left(0.001388888888888889 \cdot {im}^{6} + 1\right) \cdot \sin re} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{12} + \left(0.001388888888888889 \cdot {im}^{6} + 1\right) \cdot \sin re \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;12 + \sin re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.6e-8)
   (sin re)
   (if (<= im 1.35e+154)
     (* (+ (exp im) (exp (- im))) (* 0.5 re))
     (* 0.5 (* (sin re) (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-8) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = (exp(im) + exp(-im)) * (0.5 * re);
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.6d-8) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = (exp(im) + exp(-im)) * (0.5d0 * re)
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-8) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = (Math.exp(im) + Math.exp(-im)) * (0.5 * re);
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.6e-8:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = (math.exp(im) + math.exp(-im)) * (0.5 * re)
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.6e-8)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.6e-8)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = (exp(im) + exp(-im)) * (0.5 * re);
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.6e-8], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.5999999999999999e-8

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.7%

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

   if 5.5999999999999999e-8 < im < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 77.4%

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

      1. associate-*r* 77.4%

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

   7. Simplified 77.4%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+149}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7e+19)
   (sin re)
   (if (<= im 3.15e+149)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (* 0.5 (* (sin re) (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7e+19) {
		tmp = sin(re);
	} else if (im <= 3.15e+149) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.7d+19) then
        tmp = sin(re)
    else if (im <= 3.15d+149) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.7e+19) {
		tmp = Math.sin(re);
	} else if (im <= 3.15e+149) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.7e+19:
		tmp = math.sin(re)
	elif im <= 3.15e+149:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.7e+19)
		tmp = sin(re);
	elseif (im <= 3.15e+149)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.7e+19)
		tmp = sin(re);
	elseif (im <= 3.15e+149)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.7e+19], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.15e+149], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7e19

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.3%

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

   if 2.7e19 < im < 3.15e149

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 4.4%

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

      1. +-commutative 4.4%

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

      2. unpow2 4.4%

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

      3. fma-define 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in re around 0 12.3%

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

      1. associate-*r* 12.3%

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

      2. associate-*r* 12.3%

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

      3. *-commutative 12.3%

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

      4. distribute-rgt-out 25.2%

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

      5. +-commutative 25.2%

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

      6. unpow2 25.2%

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

      7. fma-undefine 25.2%

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

      8. *-commutative 25.2%

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

   10. Simplified 25.2%

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

   11. Taylor expanded in im around 0 21.2%

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

       1. +-commutative 21.2%

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

       2. *-commutative 21.2%

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

       3. distribute-rgt-in 21.2%

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

       4. associate-*l* 21.2%

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

       5. metadata-eval 21.2%

          \[\leadsto re \cdot \color{blue}{1} + \left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot 2 \]

       6. *-rgt-identity 21.2%

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

       7. *-commutative 21.2%

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

       8. associate-*l* 21.2%

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

       9. metadata-eval 21.2%

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

   13. Simplified 21.2%

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

   if 3.15e149 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.2%

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

      1. +-commutative 97.2%

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

      2. unpow2 97.2%

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

      3. fma-define 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in im around inf 97.2%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+149}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+117}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7e+19)
   (sin re)
   (if (<= im 2.5e+117)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (* (* 0.5 re) (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7e+19) {
		tmp = sin(re);
	} else if (im <= 2.5e+117) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.7e+19)
		tmp = sin(re);
	elseif (im <= 2.5e+117)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.7e+19], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.5e+117], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7e19

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.3%

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

   if 2.7e19 < im < 2.49999999999999992e117

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.7%

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

      1. +-commutative 3.7%

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

      2. unpow2 3.7%

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

      3. fma-define 3.7%

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

   7. Simplified 3.7%

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

   8. Taylor expanded in re around 0 11.2%

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

      1. associate-*r* 11.2%

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

      2. associate-*r* 11.2%

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

      3. *-commutative 11.2%

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

      4. distribute-rgt-out 23.7%

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

      5. +-commutative 23.7%

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

      6. unpow2 23.7%

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

      7. fma-undefine 23.7%

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

      8. *-commutative 23.7%

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

   10. Simplified 23.7%

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

   11. Taylor expanded in im around 0 22.7%

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

       1. +-commutative 22.7%

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

       2. *-commutative 22.7%

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

       3. distribute-rgt-in 22.7%

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

       4. associate-*l* 22.7%

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

       5. metadata-eval 22.7%

          \[\leadsto re \cdot \color{blue}{1} + \left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot 2 \]

       6. *-rgt-identity 22.7%

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

       7. *-commutative 22.7%

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

       8. associate-*l* 22.7%

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

       9. metadata-eval 22.7%

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

   13. Simplified 22.7%

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

   if 2.49999999999999992e117 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 75.6%

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

      1. associate-*r* 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in im around 0 66.3%

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

      1. +-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. fma-define 81.7%

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

   10. Simplified 66.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+117}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7e+19)
   (sin re)
   (if (<= im 3.8e+118)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (* re (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7e+19) {
		tmp = sin(re);
	} else if (im <= 3.8e+118) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.7d+19) then
        tmp = sin(re)
    else if (im <= 3.8d+118) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.7e+19) {
		tmp = Math.sin(re);
	} else if (im <= 3.8e+118) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.7e+19:
		tmp = math.sin(re)
	elif im <= 3.8e+118:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.7e+19)
		tmp = sin(re);
	elseif (im <= 3.8e+118)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.7e+19)
		tmp = sin(re);
	elseif (im <= 3.8e+118)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.7e+19], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.8e+118], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7e19

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.3%

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

   if 2.7e19 < im < 3.80000000000000016e118

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.7%

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

      1. +-commutative 3.7%

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

      2. unpow2 3.7%

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

      3. fma-define 3.7%

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

   7. Simplified 3.7%

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

   8. Taylor expanded in re around 0 11.2%

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

      1. associate-*r* 11.2%

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

      2. associate-*r* 11.2%

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

      3. *-commutative 11.2%

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

      4. distribute-rgt-out 23.7%

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

      5. +-commutative 23.7%

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

      6. unpow2 23.7%

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

      7. fma-undefine 23.7%

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

      8. *-commutative 23.7%

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

   10. Simplified 23.7%

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

   11. Taylor expanded in im around 0 22.7%

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

       1. +-commutative 22.7%

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

       2. *-commutative 22.7%

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

       3. distribute-rgt-in 22.7%

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

       4. associate-*l* 22.7%

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

       5. metadata-eval 22.7%

          \[\leadsto re \cdot \color{blue}{1} + \left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot 2 \]

       6. *-rgt-identity 22.7%

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

       7. *-commutative 22.7%

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

       8. associate-*l* 22.7%

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

       9. metadata-eval 22.7%

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

   13. Simplified 22.7%

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

   if 3.80000000000000016e118 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 75.6%

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

      1. associate-*r* 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in im around 0 66.3%

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

   9. Taylor expanded in im around inf 66.3%

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

       1. associate-*r* 66.3%

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

       2. *-commutative 66.3%

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

   11. Simplified 66.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.4 \cdot 10^{+36}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.4e+36) (sin re) (* re (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.4e+36) {
		tmp = sin(re);
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.4d+36) then
        tmp = sin(re)
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.4e+36) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.4e+36:
		tmp = math.sin(re)
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.4e+36)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.4e+36)
		tmp = sin(re);
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.4e+36], N[Sin[re], $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.3999999999999998e36

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 6.3999999999999998e36 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 76.7%

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

      1. associate-*r* 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in im around 0 47.8%

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

   9. Taylor expanded in im around inf 47.8%

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

       1. associate-*r* 47.8%

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

       2. *-commutative 47.8%

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

   11. Simplified 47.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.4 \cdot 10^{+36}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (sin re))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.sin(re)

Julia

function code(re, im)
	return sin(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sin[re], $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. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 52.4%

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

6. Final simplification 52.4%

   \[\leadsto \sin re \]
7. Add Preprocessing

Alternative 11: 27.0% accurate, 51.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 0.5

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 76.6%

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

      1. associate-*r* 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in im around 0 32.3%

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

   if 0.5 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 7.2%

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

4. Final simplification 25.8%

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

Alternative 12: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 2.7%

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

7. Final simplification 2.7%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 13: 4.4% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 4.6%

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

7. Final simplification 4.6%

   \[\leadsto 0.25 \]
8. Add Preprocessing

Alternative 14: 4.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 4.7%

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

7. Final simplification 4.7%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))