math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 15

Speedup: 1.0×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(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: 75.1% 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 77.8%

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

   1. neg-mul-1 77.8%

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

   2. unsub-neg 77.8%

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

7. Simplified 77.8%

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

8. Final simplification 77.8%

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

Alternative 3: 71.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.8 \lor \neg \left(im \leq 10^{+103}\right):\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 4.8) (not (<= im 1e+103)))
   (*
    0.5
    (*
     (sin re)
     (- (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))) im)))
   (* (+ (exp im) 1.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 4.8) || !(im <= 1e+103)) {
		tmp = 0.5 * (sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	} 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) .or. (.not. (im <= 1d+103))) then
        tmp = 0.5d0 * (sin(re) * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im))
    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) || !(im <= 1e+103)) {
		tmp = 0.5 * (Math.sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	} else {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 4.8) or not (im <= 1e+103):
		tmp = 0.5 * (math.sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im))
	else:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 4.8) || !(im <= 1e+103))
		tmp = Float64(0.5 * Float64(sin(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im)));
	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) || ~((im <= 1e+103)))
		tmp = 0.5 * (sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	else
		tmp = (exp(im) + 1.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 4.8], N[Not[LessEqual[im, 1e+103]], $MachinePrecision]], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $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 or 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 75.8%

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

      1. neg-mul-1 75.8%

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

      2. unsub-neg 75.8%

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

   7. Simplified 75.8%

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

   8. Taylor expanded in im around 0 74.8%

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

      1. *-commutative 74.8%

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

   10. Simplified 74.8%

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

   11. Taylor expanded in re around inf 74.8%

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

   if 4.79999999999999982 < 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 90.5%

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

      1. associate-*r* 90.5%

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

      2. *-commutative 90.5%

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

      3. +-commutative 90.5%

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

      4. associate--l+ 90.5%

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

   10. Simplified 90.5%

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

   11. Taylor expanded in im around 0 90.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8 \lor \neg \left(im \leq 10^{+103}\right):\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.6)
   (*
    (* 0.5 (sin re))
    (+ (+ 1.0 (* im (+ (* 0.5 im) -1.0))) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (if (<= im 1e+103)
     (* (+ (exp im) 1.0) (* 0.5 re))
     (*
      0.5
      (*
       (sin re)
       (-
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.6) {
		tmp = (0.5 * sin(re)) * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = 0.5 * (sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.6d0) then
        tmp = (0.5d0 * sin(re)) * ((1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1d+103) then
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    else
        tmp = 0.5d0 * (sin(re) * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.6) {
		tmp = (0.5 * Math.sin(re)) * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	} else {
		tmp = 0.5 * (Math.sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.6:
		tmp = (0.5 * math.sin(re)) * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1e+103:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	else:
		tmp = 0.5 * (math.sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.6)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0))) + 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(0.5 * Float64(sin(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.6)
		tmp = (0.5 * sin(re)) * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1e+103)
		tmp = (exp(im) + 1.0) * (0.5 * re);
	else
		tmp = 0.5 * (sin(re) * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.6], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $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[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5999999999999996

   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 91.2%

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

   6. Taylor expanded in im around 0 91.1%

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

   7. Taylor expanded in im around 0 86.8%

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

   if 4.5999999999999996 < 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 90.5%

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

      1. associate-*r* 90.5%

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

      2. *-commutative 90.5%

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

      3. +-commutative 90.5%

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

      4. associate--l+ 90.5%

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

   10. Simplified 90.5%

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

   11. Taylor expanded in im around 0 90.5%

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

   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%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around inf 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \lor \neg \left(im \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 4.6) (not (<= im 1.9e+154)))
   (* 0.5 (* (sin re) (- (+ (* im (+ 1.0 (* 0.5 im))) 2.0) im)))
   (* (+ (exp im) 1.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 4.6) || !(im <= 1.9e+154)) {
		tmp = 0.5 * (sin(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im));
	} 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 <= 4.6d0) .or. (.not. (im <= 1.9d+154))) then
        tmp = 0.5d0 * (sin(re) * (((im * (1.0d0 + (0.5d0 * im))) + 2.0d0) - im))
    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 <= 4.6) || !(im <= 1.9e+154)) {
		tmp = 0.5 * (Math.sin(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im));
	} else {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 4.6) or not (im <= 1.9e+154):
		tmp = 0.5 * (math.sin(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im))
	else:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 4.6) || !(im <= 1.9e+154))
		tmp = Float64(0.5 * Float64(sin(re) * Float64(Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0) - im)));
	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 <= 4.6) || ~((im <= 1.9e+154)))
		tmp = 0.5 * (sin(re) * (((im * (1.0 + (0.5 * im))) + 2.0) - im));
	else
		tmp = (exp(im) + 1.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 4.6], N[Not[LessEqual[im, 1.9e+154]], $MachinePrecision]], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $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 < 4.5999999999999996 or 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 74.6%

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

      1. neg-mul-1 74.6%

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

      2. unsub-neg 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in im around 0 73.6%

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

      1. *-commutative 73.6%

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

   10. Simplified 73.6%

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

   11. Taylor expanded in re around inf 73.6%

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

   12. Taylor expanded in im around 0 88.4%

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

       1. *-commutative 88.4%

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

   14. Simplified 88.4%

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

   if 4.5999999999999996 < 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 90.6%

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

      1. associate-*r* 90.6%

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

      2. *-commutative 90.6%

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

      3. +-commutative 90.6%

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

      4. associate--l+ 90.6%

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

   10. Simplified 90.6%

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

   11. Taylor expanded in im around 0 90.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \lor \neg \left(im \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.5) (sin re) (* (+ (exp im) 1.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		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 <= 3.5d0) 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 <= 3.5) {
		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 <= 3.5:
		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 <= 3.5)
		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 <= 3.5)
		tmp = sin(re);
	else
		tmp = (exp(im) + 1.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.5], 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 < 3.5

   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 70.2%

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

   if 3.5 < 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 83.9%

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

      1. associate-*r* 83.9%

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

      2. *-commutative 83.9%

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

      3. +-commutative 83.9%

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

      4. associate--l+ 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in im around 0 83.9%

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

Alternative 7: 63.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 880:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 880.0)
   (sin re)
   (*
    (* 0.5 re)
    (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 880.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * (2.0 + (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 <= 880.0d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) * (2.0d0 + (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 <= 880.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 880.0:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 880.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(2.0 + 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 <= 880.0)
		tmp = sin(re);
	else
		tmp = (0.5 * re) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 880.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 880

   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 70.2%

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

   if 880 < 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 83.9%

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

      1. associate-*r* 83.9%

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

      2. *-commutative 83.9%

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

      3. +-commutative 83.9%

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

      4. associate--l+ 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in im around 0 83.9%

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

   12. Taylor expanded in im around 0 63.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 880:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.8% accurate, 16.3× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return 0.5 * (re * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - 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 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $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 77.8%

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

   1. neg-mul-1 77.8%

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

   2. unsub-neg 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in im around 0 69.0%

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

   1. *-commutative 69.0%

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

10. Simplified 69.0%

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

11. Taylor expanded in re around 0 45.3%

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

12. Final simplification 45.3%

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

Alternative 9: 44.5% accurate, 18.2× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * re) * (2.0 + (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) * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
end function

Java

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

Python

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $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 77.8%

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

   1. neg-mul-1 77.8%

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

   2. unsub-neg 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in re around 0 51.6%

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

   1. associate-*r* 51.6%

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

   2. *-commutative 51.6%

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

   3. +-commutative 51.6%

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

   4. associate--l+ 51.6%

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

10. Simplified 51.6%

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

11. Taylor expanded in im around 0 48.5%

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

12. Taylor expanded in im around 0 45.0%

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

13. Final simplification 45.0%

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

Alternative 10: 48.8% accurate, 20.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return 0.5 * (re * (((im * (1.0 + (0.5 * im))) + 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 + (0.5d0 * im))) + 2.0d0) - im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(N[(N[(im * N[(1.0 + N[(0.5 * im), $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 77.8%

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

   1. neg-mul-1 77.8%

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

   2. unsub-neg 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in im around 0 69.0%

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

   1. *-commutative 69.0%

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

10. Simplified 69.0%

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

11. Taylor expanded in re around 0 45.3%

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

12. Taylor expanded in im around 0 53.5%

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

    1. *-commutative 77.9%

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

14. Simplified 53.5%

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

15. Final simplification 53.5%

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

Alternative 11: 48.5% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(0.5 * im), $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 77.8%

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

   1. neg-mul-1 77.8%

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

   2. unsub-neg 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in re around 0 51.6%

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

   1. associate-*r* 51.6%

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

   2. *-commutative 51.6%

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

   3. +-commutative 51.6%

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

   4. associate--l+ 51.6%

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

10. Simplified 51.6%

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

11. Taylor expanded in im around 0 48.5%

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

12. Taylor expanded in im around 0 53.2%

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

13. Final simplification 53.2%

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

Alternative 12: 30.6% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.25 \cdot 10^{+65}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot im\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.25e+65) re (* (* re im) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.25e+65) {
		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 <= 2.25d+65) 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 <= 2.25e+65) {
		tmp = re;
	} else {
		tmp = (re * im) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.25e+65:
		tmp = re
	else:
		tmp = (re * im) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.25e+65)
		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 <= 2.25e+65)
		tmp = re;
	else
		tmp = (re * im) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.25e+65], re, N[(N[(re * im), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.25e65

   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 77.7%

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

      1. +-commutative 77.7%

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

      2. unpow2 77.7%

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

      3. fma-define 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in re around 0 59.6%

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

   9. Taylor expanded in im around 0 35.9%

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

   if 2.25e65 < 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 92.6%

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

      1. neg-mul-1 92.6%

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

      2. unsub-neg 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in re around 0 32.7%

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

      1. associate-*r* 32.7%

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

      2. *-commutative 32.7%

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

      3. +-commutative 32.7%

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

      4. associate--l+ 32.7%

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

   10. Simplified 32.7%

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

   11. Taylor expanded in im around inf 10.3%

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

       1. *-commutative 10.3%

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

       2. *-commutative 10.3%

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

   13. Simplified 10.3%

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

Alternative 13: 32.9% accurate, 44.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(im + 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 77.8%

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

   1. neg-mul-1 77.8%

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

   2. unsub-neg 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in re around 0 51.6%

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

   1. associate-*r* 51.6%

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

   2. *-commutative 51.6%

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

   3. +-commutative 51.6%

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

   4. associate--l+ 51.6%

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

10. Simplified 51.6%

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

11. Taylor expanded in im around 0 48.5%

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

12. Taylor expanded in im around 0 34.3%

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

13. Final simplification 34.3%

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

Alternative 14: 27.1% 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 78.1%

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

   1. +-commutative 78.1%

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

   2. unpow2 78.1%

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

   3. fma-define 78.1%

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

7. Simplified 78.1%

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

8. Taylor expanded in re around 0 53.6%

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

9. Taylor expanded in im around 0 29.4%

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

Alternative 15: 2.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

   1. 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 3.0%

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

   1. pow-base-1 3.0%

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

   2. metadata-eval 3.0%

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

8. Simplified 3.0%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))