math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.1s

Alternatives: 18

Speedup: 1.0×

• 50.0% 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: 74.9% 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 76.3%

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

   1. neg-mul-1 76.3%

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

   2. unsub-neg 76.3%

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

7. Simplified 76.3%

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

8. Final simplification 76.3%

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

Alternative 3: 74.0% 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 76.3%

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

   1. neg-mul-1 76.3%

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

   2. unsub-neg 76.3%

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

7. Simplified 76.3%

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

8. Taylor expanded in im around 0 75.6%

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

9. Final simplification 75.6%

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

Alternative 4: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \sin re\\ t_2 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_1 \cdot \left(t\_2 + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_1 (* 0.5 (sin re)))
        (t_2
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
   (if (<= im 1.85)
     (* t_1 (+ t_2 (+ 1.0 t_0)))
     (if (<= im 1.02e+103)
       (* (* 0.5 re) (+ (exp im) t_2))
       (* t_1 (+ 2.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * sin(re);
	double t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_1 * (t_2 + (1.0 + t_0));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (exp(im) + t_2);
	} else {
		tmp = t_1 * (2.0 + t_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) :: t_2
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_1 = 0.5d0 * sin(re)
    t_2 = 1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))
    if (im <= 1.85d0) then
        tmp = t_1 * (t_2 + (1.0d0 + t_0))
    else if (im <= 1.02d+103) then
        tmp = (0.5d0 * re) * (exp(im) + t_2)
    else
        tmp = t_1 * (2.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * Math.sin(re);
	double t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_1 * (t_2 + (1.0 + t_0));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + t_2);
	} else {
		tmp = t_1 * (2.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	t_1 = 0.5 * math.sin(re)
	t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))
	tmp = 0
	if im <= 1.85:
		tmp = t_1 * (t_2 + (1.0 + t_0))
	elif im <= 1.02e+103:
		tmp = (0.5 * re) * (math.exp(im) + t_2)
	else:
		tmp = t_1 * (2.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_1 * Float64(t_2 + Float64(1.0 + t_0)));
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + t_2));
	else
		tmp = Float64(t_1 * Float64(2.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	t_1 = 0.5 * sin(re);
	t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_1 * (t_2 + (1.0 + t_0));
	elseif (im <= 1.02e+103)
		tmp = (0.5 * re) * (exp(im) + t_2);
	else
		tmp = t_1 * (2.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], N[(t$95$1 * N[(t$95$2 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.8500000000000001

   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 87.4%

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

   6. Taylor expanded in im around 0 68.9%

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

   if 1.8500000000000001 < im < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 95.8%

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

   6. Taylor expanded in re around 0 73.6%

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

      1. *-commutative 73.6%

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

   8. Simplified 73.6%

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

   if 1.01999999999999991e103 < 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re)))
        (t_1
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
   (if (<= im 1.85)
     (* t_0 (+ t_1 (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1.02e+103)
       (* (* 0.5 re) (+ (exp im) t_1))
       (*
        t_0
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (exp(im) + t_1);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    t_1 = 1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))
    if (im <= 1.85d0) then
        tmp = t_0 * (t_1 + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1.02d+103) then
        tmp = (0.5d0 * re) * (exp(im) + t_1)
    else
        tmp = t_0 * (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 t_0 = 0.5 * Math.sin(re);
	double t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + t_1);
	} else {
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))
	tmp = 0
	if im <= 1.85:
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1.02e+103:
		tmp = (0.5 * re) * (math.exp(im) + t_1)
	else:
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_0 * Float64(t_1 + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + t_1));
	else
		tmp = Float64(t_0 * 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)
	t_0 = 0.5 * sin(re);
	t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1.02e+103)
		tmp = (0.5 * re) * (exp(im) + t_1);
	else
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	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[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], N[(t$95$0 * N[(t$95$1 + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 3 regimes

2. if im < 1.8500000000000001

   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 87.4%

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

   6. Taylor expanded in im around 0 87.3%

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

      1. *-commutative 87.3%

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

   8. Simplified 87.3%

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

   if 1.8500000000000001 < im < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 95.8%

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

   6. Taylor expanded in re around 0 73.6%

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

      1. *-commutative 73.6%

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

   8. Simplified 73.6%

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

   if 1.01999999999999991e103 < 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 0.041:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + t\_1\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + t\_1\right)\\ \end{array} \end{array} \]

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 0.041)
     (* t_0 (+ (- 1.0 im) (+ 1.0 t_1)))
     (if (<= im 1.02e+103)
       (*
        (* 0.5 re)
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (* t_0 (+ 2.0 t_1))))))

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 <= 0.041) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + t_1);
	}
	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 <= 0.041d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + t_1))
    else if (im <= 1.02d+103) then
        tmp = (0.5d0 * re) * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (2.0d0 + t_1)
    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 <= 0.041) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + t_1);
	}
	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 <= 0.041:
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1))
	elif im <= 1.02e+103:
		tmp = (0.5 * re) * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (2.0 + t_1)
	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 <= 0.041)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + t_1)));
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(t_0 * Float64(2.0 + t_1));
	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 <= 0.041)
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	elseif (im <= 1.02e+103)
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (2.0 + t_1);
	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, 0.041], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0410000000000000017

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 68.5%

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

   if 0.0410000000000000017 < im < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 95.8%

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

   6. Taylor expanded in re around 0 73.6%

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

      1. *-commutative 73.6%

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

   8. Simplified 73.6%

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

   if 1.01999999999999991e103 < 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.041:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 1.85)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1.02e+103)
       (*
        (* 0.5 re)
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (*
        t_0
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= 1.85d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1.02d+103) then
        tmp = (0.5d0 * re) * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (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 t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 1.85:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1.02e+103:
		tmp = (0.5 * re) * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(t_0 * 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)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1.02e+103)
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], 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, 1.02e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 3 regimes

2. if im < 1.8500000000000001

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 83.3%

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

      1. *-commutative 87.3%

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

   10. Simplified 83.3%

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

   if 1.8500000000000001 < im < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 95.8%

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

   6. Taylor expanded in re around 0 73.6%

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

      1. *-commutative 73.6%

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

   8. Simplified 73.6%

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

   if 1.01999999999999991e103 < 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 1.85)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1.02e+103)
       (* (* 0.5 re) (+ (exp im) (+ 1.0 (* im (+ (* 0.5 im) -1.0)))))
       (*
        t_0
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= 1.85d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1.02d+103) then
        tmp = (0.5d0 * re) * (exp(im) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
    else
        tmp = t_0 * (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 t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 1.85:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1.02e+103:
		tmp = (0.5 * re) * (math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))))
	else:
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0)))));
	else
		tmp = Float64(t_0 * 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)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1.02e+103)
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	else
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], 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, 1.02e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 3 regimes

2. if im < 1.8500000000000001

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 83.3%

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

      1. *-commutative 87.3%

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

   10. Simplified 83.3%

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

   if 1.8500000000000001 < im < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 77.7%

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

      1. *-commutative 73.6%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in im around 0 73.5%

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

   if 1.01999999999999991e103 < 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85)
   (sin re)
   (if (<= im 1.02e+103)
     (* (* 0.5 re) (+ (exp im) (+ 1.0 (* im (+ (* 0.5 im) -1.0)))))
     (*
      (* 0.5 (sin re))
      (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = sin(re);
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else {
		tmp = (0.5 * sin(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 <= 1.85d0) then
        tmp = sin(re)
    else if (im <= 1.02d+103) then
        tmp = (0.5d0 * re) * (exp(im) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
    else
        tmp = (0.5d0 * sin(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 <= 1.85) {
		tmp = Math.sin(re);
	} else if (im <= 1.02e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85:
		tmp = math.sin(re)
	elif im <= 1.02e+103:
		tmp = (0.5 * re) * (math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))))
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85)
		tmp = sin(re);
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0)))));
	else
		tmp = Float64(Float64(0.5 * sin(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 <= 1.85)
		tmp = sin(re);
	elseif (im <= 1.02e+103)
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.02e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $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 3 regimes

2. if im < 1.8500000000000001

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 69.3%

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

   if 1.8500000000000001 < im < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 77.7%

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

      1. *-commutative 73.6%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in im around 0 73.5%

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

   if 1.01999999999999991e103 < 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85)
   (sin re)
   (* (* 0.5 re) (+ (exp im) (+ 1.0 (* im (+ (* 0.5 im) -1.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.85d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) * (exp(im) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * (Math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * (math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.85)
		tmp = sin(re);
	else
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8500000000000001

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 69.3%

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

   if 1.8500000000000001 < 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 re around 0 81.0%

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

      1. *-commutative 22.8%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in im around 0 79.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.82:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.82) (sin re) (* (+ (exp im) 1.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.82) {
		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 <= 1.82d0) 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 <= 1.82) {
		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 <= 1.82:
		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 <= 1.82)
		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 <= 1.82)
		tmp = sin(re);
	else
		tmp = (exp(im) + 1.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.82], 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 < 1.82000000000000006

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 69.3%

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

   if 1.82000000000000006 < 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 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in re around 0 79.7%

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

      1. associate-*r* 79.7%

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

      2. *-commutative 79.7%

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

      3. +-commutative 79.7%

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

      4. associate--l+ 79.7%

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

   10. Simplified 79.7%

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

   11. Taylor expanded in im around 0 79.7%

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

Alternative 12: 63.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85)
   (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 <= 1.85) {
		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 <= 1.85d0) 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 <= 1.85) {
		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 <= 1.85:
		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 <= 1.85)
		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 <= 1.85)
		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, 1.85], 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 < 1.8500000000000001

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in im around 0 69.3%

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

   if 1.8500000000000001 < 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 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in im around 0 98.7%

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

   9. Taylor expanded in im around 0 70.8%

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

       1. *-commutative 70.8%

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

   11. Simplified 70.8%

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

   12. Taylor expanded in re around 0 64.9%

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

       1. *-commutative 22.8%

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

   14. Simplified 64.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.9% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

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 * (0.5d0 + (im * 0.16666666666666666d0)))))) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $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 74.3%

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

6. Taylor expanded in re around 0 46.1%

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

   1. *-commutative 46.1%

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

8. Simplified 46.1%

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

9. Taylor expanded in im around 0 29.0%

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

10. Taylor expanded in im around 0 44.9%

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

11. Final simplification 44.9%

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

Alternative 14: 44.0% 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 76.3%

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

   1. neg-mul-1 76.3%

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

   2. unsub-neg 76.3%

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

7. Simplified 76.3%

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

8. Taylor expanded in re around 0 51.4%

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

   1. associate-*r* 51.4%

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

   2. *-commutative 51.4%

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

   3. +-commutative 51.4%

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

   4. associate--l+ 51.4%

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

10. Simplified 51.4%

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

11. Taylor expanded in im around 0 47.6%

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

12. Final simplification 47.6%

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

Alternative 15: 43.7% 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 76.3%

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

   1. neg-mul-1 76.3%

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

   2. unsub-neg 76.3%

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

7. Simplified 76.3%

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

8. Taylor expanded in im around 0 75.6%

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

9. Taylor expanded in im around 0 68.7%

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

    1. *-commutative 68.7%

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

11. Simplified 68.7%

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

12. Taylor expanded in re around 0 47.4%

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

    1. *-commutative 46.1%

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

14. Simplified 47.4%

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

15. Final simplification 47.4%

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

Alternative 16: 29.1% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.3 \cdot 10^{+91}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 5.3e+91) re (* re (* im -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.3e+91) {
		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 <= 5.3d+91) 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 <= 5.3e+91) {
		tmp = re;
	} else {
		tmp = re * (im * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.3e+91:
		tmp = re
	else:
		tmp = re * (im * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.3e+91)
		tmp = re;
	else
		tmp = Float64(re * Float64(im * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.3e+91)
		tmp = re;
	else
		tmp = re * (im * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.3e+91], re, N[(re * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.29999999999999997e91

   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 54.7%

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

   6. Taylor expanded in re around 0 37.0%

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

   if 5.29999999999999997e91 < 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 69.6%

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

      1. neg-mul-1 69.6%

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

      2. unsub-neg 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in re around 0 33.4%

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

      1. associate-*r* 33.4%

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

      2. *-commutative 33.4%

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

      3. +-commutative 33.4%

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

      4. associate--l+ 33.4%

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

   10. Simplified 33.4%

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

   11. Taylor expanded in im around inf 23.1%

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

       1. associate-*r* 23.1%

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

   13. Simplified 23.1%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.3 \cdot 10^{+91}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.0% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[(im + 1.0), $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 76.3%

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

   1. neg-mul-1 76.3%

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

   2. unsub-neg 76.3%

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

7. Simplified 76.3%

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

8. Taylor expanded in re around 0 51.4%

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

   1. associate-*r* 51.4%

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

   2. *-commutative 51.4%

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

   3. +-commutative 51.4%

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

   4. associate--l+ 51.4%

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

10. Simplified 51.4%

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

11. Taylor expanded in im around 0 29.4%

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

12. Taylor expanded in im around 0 34.9%

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

13. Final simplification 34.9%

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

Alternative 18: 26.3% 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 54.3%

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

6. Taylor expanded in re around 0 30.1%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))