math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 19

Speedup: 1.0×

• 48.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.0)
   (*
    (* 0.5 (sin re))
    (+
     (- 1.0 im)
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
   (if (<= im 1.15e+77)
     (* (+ (exp im) 1.0) (* 0.5 re))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.0) {
		tmp = (0.5 * sin(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else if (im <= 1.15e+77) {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.0d0) then
        tmp = (0.5d0 * sin(re)) * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    else if (im <= 1.15d+77) then
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.0) {
		tmp = (0.5 * Math.sin(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else if (im <= 1.15e+77) {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.0:
		tmp = (0.5 * math.sin(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	elif im <= 1.15e+77:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.0)
		tmp = (0.5 * sin(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	elseif (im <= 1.15e+77)
		tmp = (exp(im) + 1.0) * (0.5 * re);
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 72.4%

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

      1. neg-mul-1 72.4%

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

      2. unsub-neg 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in im around 0 71.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 71.1%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   10. Simplified 71.1%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   if 6 < 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 93.8%

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

      1. associate-*r* 93.8%

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

      2. *-commutative 93.8%

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

      3. +-commutative 93.8%

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

      4. associate--l+ 93.8%

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

   10. Simplified 93.8%

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

   11. Taylor expanded in im around 0 93.8%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. fma-define 90.9%

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

      3. associate-*r* 90.9%

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

      4. distribute-rgt-out 90.9%

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

      5. *-commutative 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp im) (- 1.0 im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(im) + (1.0 - 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) + (1.0d0 - im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(im) + (1.0 - im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(im) + (1.0 - im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + Float64(1.0 - im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(im) + (1.0 - im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Final simplification 79.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \]
9. Add Preprocessing

Alternative 4: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (sin re)) (+ (exp im) 1.0)))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(im) + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(im) + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(im) + 1.0);
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(im) + 1.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(im) + 1.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in im around 0 78.6%

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

9. Final simplification 78.6%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + 1\right) \]
10. Add Preprocessing

Alternative 5: 71.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 4.5:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + t\_1\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re)))
        (t_1 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= im 4.5)
     (* t_0 (+ (- 1.0 im) (+ 1.0 t_1)))
     (if (<= im 1e+103) (* (+ (exp im) 1.0) (* 0.5 re)) (* t_0 (+ t_1 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 4.5) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 1e+103) {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = t_0 * (t_1 + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    t_1 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (im <= 4.5d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + t_1))
    else if (im <= 1d+103) then
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    else
        tmp = t_0 * (t_1 + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 4.5) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 1e+103) {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = t_0 * (t_1 + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if im <= 4.5:
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1))
	elif im <= 1e+103:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	else:
		tmp = t_0 * (t_1 + 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (im <= 4.5)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + t_1)));
	elseif (im <= 1e+103)
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	else
		tmp = Float64(t_0 * Float64(t_1 + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (im <= 4.5)
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	elseif (im <= 1e+103)
		tmp = (exp(im) + 1.0) * (0.5 * re);
	else
		tmp = t_0 * (t_1 + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.5], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 72.4%

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

      1. neg-mul-1 72.4%

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

      2. unsub-neg 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in im around 0 71.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 71.1%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   10. Simplified 71.1%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   if 4.5 < im < 1e103

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 85.7%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

      3. +-commutative 85.7%

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

      4. associate--l+ 85.7%

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

   10. Simplified 85.7%

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

   11. Taylor expanded in im around 0 85.7%

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

   if 1e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.5:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 5.2)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1e+103)
       (* (+ (exp im) 1.0) (* 0.5 re))
       (*
        t_0
        (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 5.2) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= 5.2d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1d+103) then
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    else
        tmp = t_0 * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 5.2) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 5.2:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1e+103:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	else:
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 5.2)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1e+103)
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	else
		tmp = Float64(t_0 * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 5.2)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1e+103)
		tmp = (exp(im) + 1.0) * (0.5 * re);
	else
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5.2], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 72.4%

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

      1. neg-mul-1 72.4%

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

      2. unsub-neg 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in im around 0 85.3%

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

   if 5.20000000000000018 < im < 1e103

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 85.7%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

      3. +-commutative 85.7%

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

      4. associate--l+ 85.7%

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

   10. Simplified 85.7%

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

   11. Taylor expanded in im around 0 85.7%

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

   if 1e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4)
   (sin re)
   (if (<= im 1e+103)
     (* (+ (exp im) 1.0) (* 0.5 re))
     (*
      (* 0.5 (sin re))
      (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = sin(re);
	} else if (im <= 1e+103) {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.4d0) then
        tmp = sin(re)
    else if (im <= 1d+103) then
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = Math.sin(re);
	} else if (im <= 1e+103) {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.4:
		tmp = math.sin(re)
	elif im <= 1e+103:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.4)
		tmp = sin(re);
	elseif (im <= 1e+103)
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.4)
		tmp = sin(re);
	elseif (im <= 1e+103)
		tmp = (exp(im) + 1.0) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.4], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1e+103], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.4000000000000004

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 71.9%

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

   if 4.4000000000000004 < im < 1e103

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 85.7%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

      3. +-commutative 85.7%

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

      4. associate--l+ 85.7%

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

   10. Simplified 85.7%

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

   11. Taylor expanded in im around 0 85.7%

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

   if 1e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.4)
   (sin re)
   (if (<= im 1.9e+154)
     (* (+ (exp im) 1.0) (* 0.5 re))
     (* (* 0.5 (sin re)) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.4) {
		tmp = sin(re);
	} else if (im <= 1.9e+154) {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (0.5 * 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 <= 6.4d0) then
        tmp = sin(re)
    else if (im <= 1.9d+154) then
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.4) {
		tmp = Math.sin(re);
	} else if (im <= 1.9e+154) {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.4:
		tmp = math.sin(re)
	elif im <= 1.9e+154:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.4)
		tmp = sin(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.4)
		tmp = sin(re);
	elseif (im <= 1.9e+154)
		tmp = (exp(im) + 1.0) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.4], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.4000000000000004

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 71.9%

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

   if 6.4000000000000004 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 80.0%

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

      1. associate-*r* 80.0%

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

      2. *-commutative 80.0%

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

      3. +-commutative 80.0%

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

      4. associate--l+ 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in im around 0 80.0%

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

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.8) (sin re) (* (+ (exp im) 1.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.8) {
		tmp = sin(re);
	} else {
		tmp = (exp(im) + 1.0) * (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 <= 4.8d0) then
        tmp = sin(re)
    else
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.8) {
		tmp = Math.sin(re);
	} else {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.8:
		tmp = math.sin(re)
	else:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.8)
		tmp = sin(re);
	else
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.8)
		tmp = sin(re);
	else
		tmp = (exp(im) + 1.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.8], N[Sin[re], $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 71.9%

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

   if 4.79999999999999982 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. *-commutative 84.8%

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

      3. +-commutative 84.8%

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

      4. associate--l+ 84.8%

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

   10. Simplified 84.8%

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

   11. Taylor expanded in im around 0 84.8%

       \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 \cdot re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.02e-8)
   (sin re)
   (*
    (* 0.5 re)
    (+
     (- 1.0 im)
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 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 <= 1.02d-8) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-8) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.02e-8:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-8)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-8)
		tmp = sin(re);
	else
		tmp = (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.02e-8], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.02000000000000003e-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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 72.0%

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

   if 1.02000000000000003e-8 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in re around 0 84.3%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

      3. +-commutative 84.3%

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

      4. associate--l+ 84.3%

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

   10. Simplified 84.3%

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

   11. Taylor expanded in im around 0 64.3%

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

       1. *-commutative 69.2%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   13. Simplified 64.3%

       \[\leadsto \left(\color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} + \left(1 - im\right)\right) \cdot \left(0.5 \cdot re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 re)
  (+
   (- 1.0 im)
   (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))

C

double code(double re, double im) {
	return (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
}

Python

def code(re, im):
	return (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in re around 0 47.7%

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

   1. associate-*r* 47.7%

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

   2. *-commutative 47.7%

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

   3. +-commutative 47.7%

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

   4. associate--l+ 47.7%

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

10. Simplified 47.7%

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

11. Taylor expanded in im around 0 42.8%

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

    1. *-commutative 70.7%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

13. Simplified 42.8%

    \[\leadsto \left(\color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} + \left(1 - im\right)\right) \cdot \left(0.5 \cdot re\right) \]

14. Final simplification 42.8%

    \[\leadsto \left(0.5 \cdot re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 12: 44.1% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (*
   re
   (- (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0) im))))

C

double code(double re, double im) {
	return 0.5 * (re * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * (((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0) - im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
}

Python

def code(re, im):
	return 0.5 * (re * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0) - im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in im around 0 70.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative 70.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

10. Simplified 70.7%

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

11. Taylor expanded in re around 0 42.8%

    \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) - im\right)\right)} \]

12. Final simplification 42.8%

    \[\leadsto 0.5 \cdot \left(re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right) \]
13. Add Preprocessing

Alternative 13: 43.9% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 re)
  (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0)))

C

double code(double re, double im) {
	return (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
}

Python

def code(re, im):
	return (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in im around 0 78.6%

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

9. Taylor expanded in im around 0 70.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]

10. Taylor expanded in re around 0 42.5%

    \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

11. Final simplification 42.5%

    \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) \]
12. Add Preprocessing

Alternative 14: 47.8% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* re (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im))))

C

double code(double re, double im) {
	return 0.5 * (re * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
}

Python

def code(re, im):
	return 0.5 * (re * ((2.0 + (im * (1.0 + (0.5 * im)))) - im))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in re around 0 47.7%

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

   1. associate-*r* 47.7%

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

   2. *-commutative 47.7%

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

   3. +-commutative 47.7%

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

   4. associate--l+ 47.7%

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

10. Simplified 47.7%

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

11. Taylor expanded in im around 0 44.7%

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

12. Taylor expanded in re around 0 44.7%

    \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)} \]
13. Add Preprocessing

Alternative 15: 47.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 re) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))

C

double code(double re, double im) {
	return (0.5 * re) * (2.0 + (im * (1.0 + (0.5 * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * (2.0d0 + (im * (1.0d0 + (0.5d0 * im))))
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * (2.0 + (im * (1.0 + (0.5 * im))));
}

Python

def code(re, im):
	return (0.5 * re) * (2.0 + (im * (1.0 + (0.5 * im))))

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * (2.0 + (im * (1.0 + (0.5 * im))));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $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-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in im around 0 78.6%

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

9. Taylor expanded in im around 0 77.4%

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

10. Taylor expanded in re around 0 44.5%

    \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) \]
11. Add Preprocessing

Alternative 16: 29.5% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+107}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot im\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.05e+107) re (* (* re im) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+107) {
		tmp = re;
	} else {
		tmp = (re * im) * -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 <= 1.05d+107) then
        tmp = re
    else
        tmp = (re * im) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+107) {
		tmp = re;
	} else {
		tmp = (re * im) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e+107:
		tmp = re
	else:
		tmp = (re * im) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e+107)
		tmp = re;
	else
		tmp = Float64(Float64(re * im) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.05e+107)
		tmp = re;
	else
		tmp = (re * im) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e+107], re, N[(N[(re * im), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.05e107

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. fma-define 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in re around 0 51.8%

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

      1. +-commutative 51.8%

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

      2. unpow2 51.8%

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

      3. fma-undefine 51.8%

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

   10. Simplified 51.8%

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

   11. Taylor expanded in im around 0 27.7%

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

   if 1.05e107 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.1%

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

      1. neg-mul-1 85.1%

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

      2. unsub-neg 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in re around 0 22.2%

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

      1. associate-*r* 22.2%

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

      2. *-commutative 22.2%

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

      3. +-commutative 22.2%

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

      4. associate--l+ 22.2%

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

   10. Simplified 22.2%

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

   11. Taylor expanded in im around inf 7.4%

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

       1. *-commutative 7.4%

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

       2. *-commutative 7.4%

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

   13. Simplified 7.4%

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

Alternative 17: 32.5% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(1 + \left(1 - im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) (+ 1.0 (- 1.0 im))))

C

double code(double re, double im) {
	return (0.5 * re) * (1.0 + (1.0 - im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * (1.0d0 + (1.0d0 - im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * (1.0 + (1.0 - im));
}

Python

def code(re, im):
	return (0.5 * re) * (1.0 + (1.0 - im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(1.0 + Float64(1.0 - im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * (1.0 + (1.0 - im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[(1.0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 79.5%

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

   1. neg-mul-1 79.5%

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

   2. unsub-neg 79.5%

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

7. Simplified 79.5%

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

8. Taylor expanded in re around 0 47.7%

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

   1. associate-*r* 47.7%

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

   2. *-commutative 47.7%

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

   3. +-commutative 47.7%

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

   4. associate--l+ 47.7%

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

10. Simplified 47.7%

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

11. Taylor expanded in im around 0 28.1%

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

12. Final simplification 28.1%

    \[\leadsto \left(0.5 \cdot re\right) \cdot \left(1 + \left(1 - im\right)\right) \]
13. Add Preprocessing

Alternative 18: 26.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 78.2%

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

   1. +-commutative 78.2%

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

   2. unpow2 78.2%

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

   3. fma-define 78.2%

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

7. Simplified 78.2%

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

8. Taylor expanded in re around 0 45.3%

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

   1. +-commutative 45.3%

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

   2. unpow2 45.3%

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

   3. fma-undefine 45.3%

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

10. Simplified 45.3%

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

11. Taylor expanded in im around 0 23.5%

    \[\leadsto \color{blue}{re} \]
12. Add Preprocessing

Alternative 19: 3.0% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 2.8%

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

   1. pow-base-1 2.8%

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

   2. metadata-eval 2.8%

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

8. Simplified 2.8%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))