math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 17

Speedup: 1.0×

• 49.4% 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: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 1.02e+103)
     (*
      t_0
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (*
      t_0
      (+ 4.0 (* 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 <= 1.02e+103) {
		tmp = t_0 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (4.0 + (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 <= 1.02d+103) then
        tmp = t_0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (4.0d0 + (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 <= 1.02e+103) {
		tmp = t_0 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (4.0 + (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 <= 1.02e+103:
		tmp = t_0 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (4.0 + (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 <= 1.02e+103)
		tmp = Float64(t_0 * 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(4.0 + 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 <= 1.02e+103)
		tmp = t_0 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (4.0 + (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, 1.02e+103], N[(t$95$0 * 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[(4.0 + 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 < 1.01999999999999991e103

   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 89.9%

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

   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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 2.1:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 2.1)
     (*
      t_0
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
     (* t_0 (+ (exp im) 3.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 2.1) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = t_0 * (exp(im) + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= 2.1d0) then
        tmp = t_0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    else
        tmp = t_0 * (exp(im) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 2.1) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = t_0 * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 2.1:
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	else:
		tmp = t_0 * (math.exp(im) + 3.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 2.1)
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	else
		tmp = Float64(t_0 * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 2.1)
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	else
		tmp = t_0 * (exp(im) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.1], N[(t$95$0 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $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], N[(t$95$0 * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000009

   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 88.5%

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

   6. Taylor expanded in im around 0 67.6%

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

   if 2.10000000000000009 < 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

   6. Applied egg-rr 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{im} + 3\\ t_1 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_2 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 6:\\ \;\;\;\;t\_2 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_1\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;t\_0 \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(4 + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp im) 3.0))
        (t_1 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_2 (* 0.5 (sin re))))
   (if (<= im 6.0)
     (*
      t_2
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_1)))
     (if (<= im 1.9e+71)
       (* t_0 (* 0.5 re))
       (if (<= im 1.02e+103) (* t_0 (/ re -28.0)) (* t_2 (+ 4.0 t_1)))))))

C

double code(double re, double im) {
	double t_0 = exp(im) + 3.0;
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_2 = 0.5 * sin(re);
	double tmp;
	if (im <= 6.0) {
		tmp = t_2 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_1));
	} else if (im <= 1.9e+71) {
		tmp = t_0 * (0.5 * re);
	} else if (im <= 1.02e+103) {
		tmp = t_0 * (re / -28.0);
	} else {
		tmp = t_2 * (4.0 + 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) :: t_2
    real(8) :: tmp
    t_0 = exp(im) + 3.0d0
    t_1 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_2 = 0.5d0 * sin(re)
    if (im <= 6.0d0) then
        tmp = t_2 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_1))
    else if (im <= 1.9d+71) then
        tmp = t_0 * (0.5d0 * re)
    else if (im <= 1.02d+103) then
        tmp = t_0 * (re / (-28.0d0))
    else
        tmp = t_2 * (4.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(im) + 3.0;
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_2 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 6.0) {
		tmp = t_2 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_1));
	} else if (im <= 1.9e+71) {
		tmp = t_0 * (0.5 * re);
	} else if (im <= 1.02e+103) {
		tmp = t_0 * (re / -28.0);
	} else {
		tmp = t_2 * (4.0 + t_1);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(im) + 3.0
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	t_2 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 6.0:
		tmp = t_2 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_1))
	elif im <= 1.9e+71:
		tmp = t_0 * (0.5 * re)
	elif im <= 1.02e+103:
		tmp = t_0 * (re / -28.0)
	else:
		tmp = t_2 * (4.0 + t_1)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(im) + 3.0)
	t_1 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	t_2 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(t_2 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + t_1)));
	elseif (im <= 1.9e+71)
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (im <= 1.02e+103)
		tmp = Float64(t_0 * Float64(re / -28.0));
	else
		tmp = Float64(t_2 * Float64(4.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(im) + 3.0;
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	t_2 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 6.0)
		tmp = t_2 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_1));
	elseif (im <= 1.9e+71)
		tmp = t_0 * (0.5 * re);
	elseif (im <= 1.02e+103)
		tmp = t_0 * (re / -28.0);
	else
		tmp = t_2 * (4.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 6.0], N[(t$95$2 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+71], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+103], N[(t$95$0 * N[(re / -28.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(4.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 6

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 88.5%

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

   6. Taylor expanded in im around 0 67.6%

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

   if 6 < im < 1.9e71

   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

   6. Applied egg-rr 99.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

   9. Simplified 81.2%

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

   if 1.9e71 < im < 1.01999999999999991e103

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 33.3%

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

      1. associate-*r* 33.3%

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

      2. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \left(3 + e^{im}\right) \cdot \color{blue}{\frac{re}{-28}} \]

   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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := e^{im} + 3\\ \mathbf{if}\;im \leq 6:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+102}:\\ \;\;\;\;t\_1 \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (+ (exp im) 3.0)))
   (if (<= im 6.0)
     (*
      t_0
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1.9e+71)
       (* t_1 (* 0.5 re))
       (if (<= im 4.3e+102)
         (* t_1 (/ re -28.0))
         (*
          t_0
          (+
           4.0
           (* 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 = exp(im) + 3.0;
	double tmp;
	if (im <= 6.0) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.9e+71) {
		tmp = t_1 * (0.5 * re);
	} else if (im <= 4.3e+102) {
		tmp = t_1 * (re / -28.0);
	} else {
		tmp = t_0 * (4.0 + (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 = exp(im) + 3.0d0
    if (im <= 6.0d0) then
        tmp = t_0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1.9d+71) then
        tmp = t_1 * (0.5d0 * re)
    else if (im <= 4.3d+102) then
        tmp = t_1 * (re / (-28.0d0))
    else
        tmp = t_0 * (4.0d0 + (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 = Math.exp(im) + 3.0;
	double tmp;
	if (im <= 6.0) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.9e+71) {
		tmp = t_1 * (0.5 * re);
	} else if (im <= 4.3e+102) {
		tmp = t_1 * (re / -28.0);
	} else {
		tmp = t_0 * (4.0 + (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 = math.exp(im) + 3.0
	tmp = 0
	if im <= 6.0:
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1.9e+71:
		tmp = t_1 * (0.5 * re)
	elif im <= 4.3e+102:
		tmp = t_1 * (re / -28.0)
	else:
		tmp = t_0 * (4.0 + (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(exp(im) + 3.0)
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1.9e+71)
		tmp = Float64(t_1 * Float64(0.5 * re));
	elseif (im <= 4.3e+102)
		tmp = Float64(t_1 * Float64(re / -28.0));
	else
		tmp = Float64(t_0 * Float64(4.0 + 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 = exp(im) + 3.0;
	tmp = 0.0;
	if (im <= 6.0)
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1.9e+71)
		tmp = t_1 * (0.5 * re);
	elseif (im <= 4.3e+102)
		tmp = t_1 * (re / -28.0);
	else
		tmp = t_0 * (4.0 + (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[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[im, 6.0], N[(t$95$0 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+71], N[(t$95$1 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.3e+102], N[(t$95$1 * N[(re / -28.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(4.0 + 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 4 regimes

2. if im < 6

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 88.5%

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

   6. Taylor expanded in im around 0 88.3%

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

   if 6 < im < 1.9e71

   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

   6. Applied egg-rr 99.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

   9. Simplified 81.2%

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

   if 1.9e71 < im < 4.3000000000000001e102

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 33.3%

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

      1. associate-*r* 33.3%

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

      2. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \left(3 + e^{im}\right) \cdot \color{blue}{\frac{re}{-28}} \]

   if 4.3000000000000001e102 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;im \leq 0.0058:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + t\_1\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

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 0.0058)
     (* t_0 (+ t_1 (+ im 1.0)))
     (if (<= im 1.9e+71)
       (* (+ (exp im) t_1) (* 0.5 re))
       (if (<= im 1.75e+98)
         (* (+ (exp im) 3.0) (/ re -28.0))
         (*
          t_0
          (+
           4.0
           (* 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 <= 0.0058) {
		tmp = t_0 * (t_1 + (im + 1.0));
	} else if (im <= 1.9e+71) {
		tmp = (exp(im) + t_1) * (0.5 * re);
	} else if (im <= 1.75e+98) {
		tmp = (exp(im) + 3.0) * (re / -28.0);
	} else {
		tmp = t_0 * (4.0 + (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 <= 0.0058d0) then
        tmp = t_0 * (t_1 + (im + 1.0d0))
    else if (im <= 1.9d+71) then
        tmp = (exp(im) + t_1) * (0.5d0 * re)
    else if (im <= 1.75d+98) then
        tmp = (exp(im) + 3.0d0) * (re / (-28.0d0))
    else
        tmp = t_0 * (4.0d0 + (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 <= 0.0058) {
		tmp = t_0 * (t_1 + (im + 1.0));
	} else if (im <= 1.9e+71) {
		tmp = (Math.exp(im) + t_1) * (0.5 * re);
	} else if (im <= 1.75e+98) {
		tmp = (Math.exp(im) + 3.0) * (re / -28.0);
	} else {
		tmp = t_0 * (4.0 + (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 <= 0.0058:
		tmp = t_0 * (t_1 + (im + 1.0))
	elif im <= 1.9e+71:
		tmp = (math.exp(im) + t_1) * (0.5 * re)
	elif im <= 1.75e+98:
		tmp = (math.exp(im) + 3.0) * (re / -28.0)
	else:
		tmp = t_0 * (4.0 + (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 <= 0.0058)
		tmp = Float64(t_0 * Float64(t_1 + Float64(im + 1.0)));
	elseif (im <= 1.9e+71)
		tmp = Float64(Float64(exp(im) + t_1) * Float64(0.5 * re));
	elseif (im <= 1.75e+98)
		tmp = Float64(Float64(exp(im) + 3.0) * Float64(re / -28.0));
	else
		tmp = Float64(t_0 * Float64(4.0 + 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 <= 0.0058)
		tmp = t_0 * (t_1 + (im + 1.0));
	elseif (im <= 1.9e+71)
		tmp = (exp(im) + t_1) * (0.5 * re);
	elseif (im <= 1.75e+98)
		tmp = (exp(im) + 3.0) * (re / -28.0);
	else
		tmp = t_0 * (4.0 + (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, 0.0058], N[(t$95$0 * N[(t$95$1 + N[(im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+71], N[(N[(N[Exp[im], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.75e+98], N[(N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision] * N[(re / -28.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(4.0 + 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 4 regimes

2. if im < 0.0058

   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 88.6%

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

   6. Taylor expanded in im around 0 88.2%

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

   if 0.0058 < im < 1.9e71

   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 98.2%

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

   6. Taylor expanded in re around 0 80.6%

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

   if 1.9e71 < im < 1.75e98

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 33.3%

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

      1. associate-*r* 33.3%

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

      2. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \left(3 + e^{im}\right) \cdot \color{blue}{\frac{re}{-28}} \]

   if 1.75e98 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0058:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00305:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00305)
   (sin re)
   (if (<= im 1.9e+71)
     (*
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0))))
      (* 0.5 re))
     (if (<= im 1.02e+103)
       (* (+ (exp im) 3.0) (/ re -28.0))
       (*
        (* 0.5 (sin re))
        (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00305) {
		tmp = sin(re);
	} else if (im <= 1.9e+71) {
		tmp = (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0)))) * (0.5 * re);
	} else if (im <= 1.02e+103) {
		tmp = (exp(im) + 3.0) * (re / -28.0);
	} else {
		tmp = (0.5 * sin(re)) * (4.0 + (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 <= 0.00305d0) then
        tmp = sin(re)
    else if (im <= 1.9d+71) then
        tmp = (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0))))) * (0.5d0 * re)
    else if (im <= 1.02d+103) then
        tmp = (exp(im) + 3.0d0) * (re / (-28.0d0))
    else
        tmp = (0.5d0 * sin(re)) * (4.0d0 + (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 <= 0.00305) {
		tmp = Math.sin(re);
	} else if (im <= 1.9e+71) {
		tmp = (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0)))) * (0.5 * re);
	} else if (im <= 1.02e+103) {
		tmp = (Math.exp(im) + 3.0) * (re / -28.0);
	} else {
		tmp = (0.5 * Math.sin(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00305:
		tmp = math.sin(re)
	elif im <= 1.9e+71:
		tmp = (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0)))) * (0.5 * re)
	elif im <= 1.02e+103:
		tmp = (math.exp(im) + 3.0) * (re / -28.0)
	else:
		tmp = (0.5 * math.sin(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00305)
		tmp = sin(re);
	elseif (im <= 1.9e+71)
		tmp = Float64(Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))) * Float64(0.5 * re));
	elseif (im <= 1.02e+103)
		tmp = Float64(Float64(exp(im) + 3.0) * Float64(re / -28.0));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(4.0 + 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 <= 0.00305)
		tmp = sin(re);
	elseif (im <= 1.9e+71)
		tmp = (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0)))) * (0.5 * re);
	elseif (im <= 1.02e+103)
		tmp = (exp(im) + 3.0) * (re / -28.0);
	else
		tmp = (0.5 * sin(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00305], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.9e+71], N[(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] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+103], N[(N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision] * N[(re / -28.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + 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 4 regimes

2. if im < 0.00305000000000000019

   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 67.8%

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

   if 0.00305000000000000019 < im < 1.9e71

   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 98.2%

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

   6. Taylor expanded in re around 0 80.6%

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

   if 1.9e71 < im < 1.01999999999999991e103

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 33.3%

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

      1. associate-*r* 33.3%

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

      2. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \left(3 + e^{im}\right) \cdot \color{blue}{\frac{re}{-28}} \]

   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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00305:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{im} + 3\\ \mathbf{if}\;im \leq 6:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+102}:\\ \;\;\;\;t\_0 \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp im) 3.0)))
   (if (<= im 6.0)
     (sin re)
     (if (<= im 1.9e+71)
       (* t_0 (* 0.5 re))
       (if (<= im 8.5e+102)
         (* t_0 (/ re -28.0))
         (*
          (* 0.5 (sin re))
          (+
           4.0
           (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))))

C

double code(double re, double im) {
	double t_0 = exp(im) + 3.0;
	double tmp;
	if (im <= 6.0) {
		tmp = sin(re);
	} else if (im <= 1.9e+71) {
		tmp = t_0 * (0.5 * re);
	} else if (im <= 8.5e+102) {
		tmp = t_0 * (re / -28.0);
	} else {
		tmp = (0.5 * sin(re)) * (4.0 + (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 = exp(im) + 3.0d0
    if (im <= 6.0d0) then
        tmp = sin(re)
    else if (im <= 1.9d+71) then
        tmp = t_0 * (0.5d0 * re)
    else if (im <= 8.5d+102) then
        tmp = t_0 * (re / (-28.0d0))
    else
        tmp = (0.5d0 * sin(re)) * (4.0d0 + (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 = Math.exp(im) + 3.0;
	double tmp;
	if (im <= 6.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.9e+71) {
		tmp = t_0 * (0.5 * re);
	} else if (im <= 8.5e+102) {
		tmp = t_0 * (re / -28.0);
	} else {
		tmp = (0.5 * Math.sin(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(im) + 3.0
	tmp = 0
	if im <= 6.0:
		tmp = math.sin(re)
	elif im <= 1.9e+71:
		tmp = t_0 * (0.5 * re)
	elif im <= 8.5e+102:
		tmp = t_0 * (re / -28.0)
	else:
		tmp = (0.5 * math.sin(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(im) + 3.0)
	tmp = 0.0
	if (im <= 6.0)
		tmp = sin(re);
	elseif (im <= 1.9e+71)
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (im <= 8.5e+102)
		tmp = Float64(t_0 * Float64(re / -28.0));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(4.0 + 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 = exp(im) + 3.0;
	tmp = 0.0;
	if (im <= 6.0)
		tmp = sin(re);
	elseif (im <= 1.9e+71)
		tmp = t_0 * (0.5 * re);
	elseif (im <= 8.5e+102)
		tmp = t_0 * (re / -28.0);
	else
		tmp = (0.5 * sin(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[im, 6.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.9e+71], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+102], N[(t$95$0 * N[(re / -28.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + 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 4 regimes

2. if im < 6

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 6 < im < 1.9e71

   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

   6. Applied egg-rr 99.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

   9. Simplified 81.2%

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

   if 1.9e71 < im < 8.4999999999999996e102

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 33.3%

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

      1. associate-*r* 33.3%

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

      2. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \left(3 + e^{im}\right) \cdot \color{blue}{\frac{re}{-28}} \]

   if 8.4999999999999996e102 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \frac{re}{-28}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2)
   (sin re)
   (if (<= im 1.02e+154)
     (* (+ (exp im) 3.0) (* 0.5 re))
     (* (* 0.5 (sin re)) (+ 4.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2) {
		tmp = sin(re);
	} else if (im <= 1.02e+154) {
		tmp = (exp(im) + 3.0) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (4.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.2d0) then
        tmp = sin(re)
    else if (im <= 1.02d+154) then
        tmp = (exp(im) + 3.0d0) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (4.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.2) {
		tmp = Math.sin(re);
	} else if (im <= 1.02e+154) {
		tmp = (Math.exp(im) + 3.0) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (4.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.2:
		tmp = math.sin(re)
	elif im <= 1.02e+154:
		tmp = (math.exp(im) + 3.0) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (4.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2)
		tmp = sin(re);
	elseif (im <= 1.02e+154)
		tmp = Float64(Float64(exp(im) + 3.0) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(4.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.2)
		tmp = sin(re);
	elseif (im <= 1.02e+154)
		tmp = (exp(im) + 3.0) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (4.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.02e+154], N[(N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.20000000000000018

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 4.20000000000000018 < im < 1.02000000000000007e154

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 58.8%

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

      1. associate-*r* 58.8%

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

      2. *-commutative 58.8%

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

   9. Simplified 58.8%

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

   if 1.02000000000000007e154 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3) (sin re) (* (+ (exp im) 3.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = sin(re);
	} else {
		tmp = (exp(im) + 3.0) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.3d0) then
        tmp = sin(re)
    else
        tmp = (exp(im) + 3.0d0) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = Math.sin(re);
	} else {
		tmp = (Math.exp(im) + 3.0) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.3:
		tmp = math.sin(re)
	else:
		tmp = (math.exp(im) + 3.0) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.3)
		tmp = sin(re);
	else
		tmp = Float64(Float64(exp(im) + 3.0) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.3)
		tmp = sin(re);
	else
		tmp = (exp(im) + 3.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.3], N[Sin[re], $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2999999999999998

   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 67.6%

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

   if 3.2999999999999998 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 15500000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re + 29\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 15500000.0)
   (sin re)
   (*
    (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
    (+ re 29.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 15500000.0) {
		tmp = sin(re);
	} else {
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re + 29.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 15500000.0d0) then
        tmp = sin(re)
    else
        tmp = (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) * (re + 29.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 15500000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re + 29.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 15500000.0:
		tmp = math.sin(re)
	else:
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re + 29.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 15500000.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) * Float64(re + 29.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 15500000.0)
		tmp = sin(re);
	else
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re + 29.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 15500000.0], N[Sin[re], $MachinePrecision], N[(N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 29.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.55e7

   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 67.0%

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

   if 1.55e7 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 71.4%

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

      1. associate-*r* 71.4%

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

      2. *-commutative 71.4%

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

   9. Simplified 71.4%

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

   10. Taylor expanded in im around 0 54.9%

       \[\leadsto \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \cdot \left(0.5 \cdot re\right) \]
   11. Step-by-step derivation

       1. *-commutative 68.7%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   12. Simplified 54.9%

       \[\leadsto \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(0.5 \cdot re\right) \]

   14. Applied egg-rr 24.9%

       \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - -28\right)} \]
   15. Step-by-step derivation

       1. log1p-undefine 24.9%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + re\right)}} - -28\right) \]

       2. rem-exp-log 37.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\color{blue}{\left(1 + re\right)} - -28\right) \]

       3. +-commutative 37.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\color{blue}{\left(re + 1\right)} - -28\right) \]

       4. associate--l+ 37.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(re + \left(1 - -28\right)\right)} \]

       5. metadata-eval 37.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re + \color{blue}{29}\right) \]

   16. Simplified 37.8%

       \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(re + 29\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.05 \cdot 10^{+27} \lor \neg \left(re \leq 3.65 \cdot 10^{+261}\right):\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 3.05e+27) (not (<= re 3.65e+261)))
   (+ re (* 0.5 (* re (* im im))))
   (*
    (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
    (* re -27.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 3.05e+27) || !(re <= 3.65e+261)) {
		tmp = re + (0.5 * (re * (im * im)));
	} else {
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re * -27.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 3.05d+27) .or. (.not. (re <= 3.65d+261))) then
        tmp = re + (0.5d0 * (re * (im * im)))
    else
        tmp = (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) * (re * (-27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 3.05e+27) || !(re <= 3.65e+261)) {
		tmp = re + (0.5 * (re * (im * im)));
	} else {
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re * -27.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 3.05e+27) or not (re <= 3.65e+261):
		tmp = re + (0.5 * (re * (im * im)))
	else:
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re * -27.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 3.05e+27) || !(re <= 3.65e+261))
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	else
		tmp = Float64(Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) * Float64(re * -27.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 3.05e+27) || ~((re <= 3.65e+261)))
		tmp = re + (0.5 * (re * (im * im)));
	else
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (re * -27.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 3.05e+27], N[Not[LessEqual[re, 3.65e+261]], $MachinePrecision]], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * -27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.0499999999999999e27 or 3.6499999999999998e261 < 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 re around 0 72.8%

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

   6. Taylor expanded in im around 0 57.9%

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

      1. unpow2 57.9%

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

   8. Applied egg-rr 57.9%

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

   if 3.0499999999999999e27 < re < 3.6499999999999998e261

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 33.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 7.0%

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

      1. associate-*r* 7.0%

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

      2. *-commutative 7.0%

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

   9. Simplified 7.0%

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

   10. Taylor expanded in im around 0 13.1%

       \[\leadsto \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \cdot \left(0.5 \cdot re\right) \]
   11. Step-by-step derivation

       1. *-commutative 28.7%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   12. Simplified 13.1%

       \[\leadsto \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(0.5 \cdot re\right) \]

   14. Applied egg-rr 23.8%

       \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, -28, re\right)} \]
   15. Step-by-step derivation

       1. fma-undefine 23.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(re \cdot -28 + re\right)} \]

       2. *-commutative 23.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\color{blue}{-28 \cdot re} + re\right) \]

       3. distribute-lft1-in 23.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(\left(-28 + 1\right) \cdot re\right)} \]

       4. *-commutative 23.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(re \cdot \left(-28 + 1\right)\right)} \]

       5. metadata-eval 23.8%

          \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \color{blue}{-27}\right) \]

   16. Simplified 23.8%

       \[\leadsto \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(re \cdot -27\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.05 \cdot 10^{+27} \lor \neg \left(re \leq 3.65 \cdot 10^{+261}\right):\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.0)
   (+ re (* 0.5 (* re (* im im))))
   (*
    (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
    (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.0) {
		tmp = re + (0.5 * (re * (im * im)));
	} else {
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (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 <= 2.0d0) then
        tmp = re + (0.5d0 * (re * (im * im)))
    else
        tmp = (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.0) {
		tmp = re + (0.5 * (re * (im * im)));
	} else {
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.0:
		tmp = re + (0.5 * (re * (im * im)))
	else:
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.0)
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	else
		tmp = Float64(Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.0)
		tmp = re + (0.5 * (re * (im * im)));
	else
		tmp = (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.0], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2

   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 59.7%

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

   6. Taylor expanded in im around 0 49.1%

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

      1. unpow2 49.1%

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

   8. Applied egg-rr 49.1%

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

   if 2 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

   9. Simplified 70.8%

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

   10. Taylor expanded in im around 0 53.5%

       \[\leadsto \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \cdot \left(0.5 \cdot re\right) \]
   11. Step-by-step derivation

       1. *-commutative 67.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   12. Simplified 53.5%

       \[\leadsto \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(0.5 \cdot re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.8% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.05:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.05) re (* (* 0.5 re) (+ im 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.05) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im + 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.05d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * (im + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.05) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.05:
		tmp = re
	else:
		tmp = (0.5 * re) * (im + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.05)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.05)
		tmp = re;
	else
		tmp = (0.5 * re) * (im + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.05], re, N[(N[(0.5 * re), $MachinePrecision] * N[(im + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.05000000000000004

   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 59.7%

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

   6. Taylor expanded in im around 0 35.6%

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

   if 1.05000000000000004 < 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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

   9. Simplified 70.8%

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

   10. Taylor expanded in im around 0 11.0%

       \[\leadsto \color{blue}{\left(4 + im\right)} \cdot \left(0.5 \cdot re\right) \]
   11. Step-by-step derivation

       1. +-commutative 11.0%

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

   12. Simplified 11.0%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.05:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.1% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-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 62.8%

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

6. Taylor expanded in im around 0 49.2%

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

   1. unpow2 49.2%

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

8. Applied egg-rr 49.2%

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

9. Final simplification 49.2%

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

Alternative 16: 27.6% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.0) re 1.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = re;
	} else {
		tmp = 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 (re <= 1.0d0) then
        tmp = re
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = re;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.0:
		tmp = re
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.0)
		tmp = re;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.0)
		tmp = re;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.0], re, 1.0]

Derivation

1. Split input into 2 regimes

2. if re < 1

   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 74.6%

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

   6. Taylor expanded in im around 0 33.1%

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

   if 1 < 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 72.8%

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

      1. +-commutative 72.8%

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

      2. unpow2 72.8%

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

      3. fma-define 72.8%

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

   7. Simplified 72.8%

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

   9. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{\frac{\sin re}{\sin re + \left(\sin re - \sin re\right)}} \]
   10. Step-by-step derivation

       1. +-inverses 6.7%

          \[\leadsto \frac{\sin re}{\sin re + \color{blue}{0}} \]

       2. +-rgt-identity 6.7%

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

       3. *-inverses 6.7%

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

   11. Simplified 6.7%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 4.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. distribute-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 76.1%

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

   1. +-commutative 76.1%

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

   2. unpow2 76.1%

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

   3. fma-define 76.1%

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

7. Simplified 76.1%

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

9. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{\frac{\sin re}{\sin re + \left(\sin re - \sin re\right)}} \]
10. Step-by-step derivation

    1. +-inverses 4.4%

       \[\leadsto \frac{\sin re}{\sin re + \color{blue}{0}} \]

    2. +-rgt-identity 4.4%

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

    3. *-inverses 4.4%

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

11. Simplified 4.4%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024188 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))