math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 18

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 74.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.5%

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

6. Final simplification 77.5%

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

Alternative 3: 72.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6e-5)
   (sin re)
   (if (<= im 1.05e+103)
     (*
      (* 0.5 re)
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (*
      (* 0.5 (sin re))
      (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6e-5) {
		tmp = sin(re);
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6d-5) then
        tmp = sin(re)
    else if (im <= 1.05d+103) then
        tmp = (0.5d0 * re) * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6e-5) {
		tmp = Math.sin(re);
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6e-5:
		tmp = math.sin(re)
	elif im <= 1.05e+103:
		tmp = (0.5 * re) * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6e-5)
		tmp = sin(re);
	elseif (im <= 1.05e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6e-5)
		tmp = sin(re);
	elseif (im <= 1.05e+103)
		tmp = (0.5 * re) * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6e-5], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.00000000000000015e-5

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 6.00000000000000015e-5 < im < 1.0500000000000001e103

   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 77.3%

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

   6. Taylor expanded in im around 0 77.3%

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

   if 1.0500000000000001e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8)
   (sin re)
   (if (<= im 1.05e+103)
     (* (* 0.5 re) (+ (exp im) 3.0))
     (*
      (* 0.5 (sin re))
      (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = sin(re);
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * re) * (exp(im) + 3.0);
	} else {
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.8d0) then
        tmp = sin(re)
    else if (im <= 1.05d+103) then
        tmp = (0.5d0 * re) * (exp(im) + 3.0d0)
    else
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = Math.sin(re);
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * re) * (Math.exp(im) + 3.0);
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = math.sin(re)
	elif im <= 1.05e+103:
		tmp = (0.5 * re) * (math.exp(im) + 3.0)
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		tmp = sin(re);
	elseif (im <= 1.05e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + 3.0));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8)
		tmp = sin(re);
	elseif (im <= 1.05e+103)
		tmp = (0.5 * re) * (exp(im) + 3.0);
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 7.79999999999999982 < im < 1.0500000000000001e103

   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 77.3%

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

   7. Applied egg-rr 77.3%

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

   if 1.0500000000000001e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 2\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.0)
   (sin re)
   (if (<= im 1.6e+151)
     (* (* 0.5 re) (+ (exp im) 3.0))
     (* (* (sin re) 2.0) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = sin(re);
	} else if (im <= 1.6e+151) {
		tmp = (0.5 * re) * (exp(im) + 3.0);
	} else {
		tmp = (sin(re) * 2.0) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8.0d0) then
        tmp = sin(re)
    else if (im <= 1.6d+151) then
        tmp = (0.5d0 * re) * (exp(im) + 3.0d0)
    else
        tmp = (sin(re) * 2.0d0) * (2.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.6e+151) {
		tmp = (0.5 * re) * (Math.exp(im) + 3.0);
	} else {
		tmp = (Math.sin(re) * 2.0) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.0:
		tmp = math.sin(re)
	elif im <= 1.6e+151:
		tmp = (0.5 * re) * (math.exp(im) + 3.0)
	else:
		tmp = (math.sin(re) * 2.0) * (2.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.0)
		tmp = sin(re);
	elseif (im <= 1.6e+151)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + 3.0));
	else
		tmp = Float64(Float64(sin(re) * 2.0) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.0)
		tmp = sin(re);
	elseif (im <= 1.6e+151)
		tmp = (0.5 * re) * (exp(im) + 3.0);
	else
		tmp = (sin(re) * 2.0) * (2.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.6e+151], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * 2.0), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 8 < im < 1.59999999999999997e151

   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 80.0%

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

   7. Applied egg-rr 80.0%

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

   if 1.59999999999999997e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. count-2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 2\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8)
   (sin re)
   (if (<= im 1.6e+151)
     (* (* 0.5 re) (+ (exp im) 3.0))
     (* (* 0.5 (sin re)) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = sin(re);
	} else if (im <= 1.6e+151) {
		tmp = (0.5 * re) * (exp(im) + 3.0);
	} else {
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.8d0) then
        tmp = sin(re)
    else if (im <= 1.6d+151) then
        tmp = (0.5d0 * re) * (exp(im) + 3.0d0)
    else
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = Math.sin(re);
	} else if (im <= 1.6e+151) {
		tmp = (0.5 * re) * (Math.exp(im) + 3.0);
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = math.sin(re)
	elif im <= 1.6e+151:
		tmp = (0.5 * re) * (math.exp(im) + 3.0)
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		tmp = sin(re);
	elseif (im <= 1.6e+151)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + 3.0));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8)
		tmp = sin(re);
	elseif (im <= 1.6e+151)
		tmp = (0.5 * re) * (exp(im) + 3.0);
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.6e+151], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 7.79999999999999982 < im < 1.59999999999999997e151

   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 80.0%

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

   7. Applied egg-rr 80.0%

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

   if 1.59999999999999997e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+75}:\\ \;\;\;\;{re}^{-2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(re + 3\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 850.0)
   (sin re)
   (if (<= im 4.1e+75)
     (* (pow re -2.0) 0.25)
     (*
      (+ re 3.0)
      (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = sin(re);
	} else if (im <= 4.1e+75) {
		tmp = pow(re, -2.0) * 0.25;
	} else {
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 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 <= 850.0d0) then
        tmp = sin(re)
    else if (im <= 4.1d+75) then
        tmp = (re ** (-2.0d0)) * 0.25d0
    else
        tmp = (re + 3.0d0) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.1e+75) {
		tmp = Math.pow(re, -2.0) * 0.25;
	} else {
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 850.0:
		tmp = math.sin(re)
	elif im <= 4.1e+75:
		tmp = math.pow(re, -2.0) * 0.25
	else:
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 850.0)
		tmp = sin(re);
	elseif (im <= 4.1e+75)
		tmp = Float64((re ^ -2.0) * 0.25);
	else
		tmp = Float64(Float64(re + 3.0) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 850.0)
		tmp = sin(re);
	elseif (im <= 4.1e+75)
		tmp = (re ^ -2.0) * 0.25;
	else
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 850.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.1e+75], N[(N[Power[re, -2.0], $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(re + 3.0), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 850

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 850 < im < 4.0999999999999998e75

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 81.3%

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

   7. Applied egg-rr 2.4%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{0.25} \]

   9. Applied egg-rr 8.6%

      \[\leadsto \color{blue}{{re}^{-2}} \cdot 0.25 \]

   if 4.0999999999999998e75 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 72.3%

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

   7. Applied egg-rr 72.3%

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

   8. Taylor expanded in im around 0 68.3%

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

   10. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - -2\right)} \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]
   11. Step-by-step derivation

       1. log1p-undefine 55.3%

          \[\leadsto \left(e^{\color{blue}{\log \left(1 + re\right)}} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       2. rem-exp-log 59.6%

          \[\leadsto \left(\color{blue}{\left(1 + re\right)} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       3. +-commutative 59.6%

          \[\leadsto \left(\color{blue}{\left(re + 1\right)} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       4. associate--l+ 59.6%

          \[\leadsto \color{blue}{\left(re + \left(1 - -2\right)\right)} \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       5. metadata-eval 59.6%

          \[\leadsto \left(re + \color{blue}{3}\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

   12. Simplified 59.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+75}:\\ \;\;\;\;{re}^{-2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(re + 3\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8) (sin re) (* (* 0.5 re) (+ (exp im) 3.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * (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) :: tmp
    if (im <= 7.8d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) * (exp(im) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * (math.exp(im) + 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8)
		tmp = sin(re);
	else
		tmp = (0.5 * re) * (exp(im) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 7.79999999999999982 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10.5:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 10.5) (sin re) (* re (+ (exp im) 3.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 10.5) {
		tmp = sin(re);
	} else {
		tmp = re * (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) :: tmp
    if (im <= 10.5d0) then
        tmp = sin(re)
    else
        tmp = re * (exp(im) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 10.5) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 10.5:
		tmp = math.sin(re)
	else:
		tmp = re * (math.exp(im) + 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 10.5)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 10.5)
		tmp = sin(re);
	else
		tmp = re * (exp(im) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 10.5], N[Sin[re], $MachinePrecision], N[(re * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 10.5

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 10.5 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

   9. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(3 + e^{im}\right) \]
   10. Step-by-step derivation

       1. fma-undefine 25.4%

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

       2. distribute-lft1-in 25.4%

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

       3. metadata-eval 25.4%

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

       4. neg-mul-1 25.4%

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

   11. Simplified 25.4%

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

       1. add-sqr-sqrt 7.9%

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

       2. sqrt-unprod 44.4%

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

       3. sqr-neg 44.4%

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

       4. sqrt-unprod 49.2%

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

       5. add-sqr-sqrt 74.6%

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

       6. distribute-lft-in 74.6%

          \[\leadsto \color{blue}{re \cdot 3 + re \cdot e^{im}} \]

   13. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{re \cdot 3 + re \cdot e^{im}} \]
   14. Step-by-step derivation

       1. distribute-lft-in 74.6%

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

   15. Simplified 74.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10.5:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re + 3\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.1e+16)
   (sin re)
   (*
    (+ re 3.0)
    (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+16) {
		tmp = sin(re);
	} else {
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 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 <= 1.1d+16) then
        tmp = sin(re)
    else
        tmp = (re + 3.0d0) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+16) {
		tmp = Math.sin(re);
	} else {
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.1e+16:
		tmp = math.sin(re)
	else:
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.1e+16)
		tmp = sin(re);
	else
		tmp = Float64(Float64(re + 3.0) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.1e+16)
		tmp = sin(re);
	else
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.1e+16], N[Sin[re], $MachinePrecision], N[(N[(re + 3.0), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1e16

   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 68.7%

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

   if 1.1e16 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 75.4%

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in im around 0 58.5%

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

   10. Applied egg-rr 45.9%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - -2\right)} \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]
   11. Step-by-step derivation

       1. log1p-undefine 45.9%

          \[\leadsto \left(e^{\color{blue}{\log \left(1 + re\right)}} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       2. rem-exp-log 51.2%

          \[\leadsto \left(\color{blue}{\left(1 + re\right)} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       3. +-commutative 51.2%

          \[\leadsto \left(\color{blue}{\left(re + 1\right)} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       4. associate--l+ 51.2%

          \[\leadsto \color{blue}{\left(re + \left(1 - -2\right)\right)} \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       5. metadata-eval 51.2%

          \[\leadsto \left(re + \color{blue}{3}\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

   12. Simplified 51.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re + 3\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re + 3\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125.0)
   re
   (*
    (+ re 3.0)
    (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = re;
	} else {
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 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 <= 125.0d0) then
        tmp = re
    else
        tmp = (re + 3.0d0) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = re;
	} else {
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 125.0:
		tmp = re
	else:
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 125.0)
		tmp = re;
	else
		tmp = Float64(Float64(re + 3.0) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 125.0)
		tmp = re;
	else
		tmp = (re + 3.0) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 125.0], re, N[(N[(re + 3.0), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 125

   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 62.4%

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

   6. Taylor expanded in im around 0 40.9%

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

   if 125 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in im around 0 53.1%

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

   10. Applied egg-rr 41.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - -2\right)} \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]
   11. Step-by-step derivation

       1. log1p-undefine 41.6%

          \[\leadsto \left(e^{\color{blue}{\log \left(1 + re\right)}} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       2. rem-exp-log 46.5%

          \[\leadsto \left(\color{blue}{\left(1 + re\right)} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       3. +-commutative 46.5%

          \[\leadsto \left(\color{blue}{\left(re + 1\right)} - -2\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       4. associate--l+ 46.5%

          \[\leadsto \color{blue}{\left(re + \left(1 - -2\right)\right)} \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

       5. metadata-eval 46.5%

          \[\leadsto \left(re + \color{blue}{3}\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right) \]

   12. Simplified 46.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re + 3\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8)
   re
   (*
    (* 0.5 re)
    (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 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 <= 7.8d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = re
	else:
		tmp = (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8)
		tmp = re;
	else
		tmp = (0.5 * re) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], re, N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 62.4%

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

   6. Taylor expanded in im around 0 40.9%

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

   if 7.79999999999999982 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in im around 0 53.1%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.0% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8)
   re
   (* (* 0.5 re) (+ 4.0 (* im (+ 1.0 (* im (* im 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (4.0 + (im * (1.0 + (im * (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 <= 7.8d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * (4.0d0 + (im * (1.0d0 + (im * (im * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (4.0 + (im * (1.0 + (im * (im * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = re
	else:
		tmp = (0.5 * re) * (4.0 + (im * (1.0 + (im * (im * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(im * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8)
		tmp = re;
	else
		tmp = (0.5 * re) * (4.0 + (im * (1.0 + (im * (im * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], re, N[(N[(0.5 * re), $MachinePrecision] * N[(4.0 + N[(im * N[(1.0 + N[(im * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 62.4%

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

   6. Taylor expanded in im around 0 40.9%

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

   if 7.79999999999999982 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in im around 0 53.1%

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

   9. Taylor expanded in im around inf 53.1%

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

       1. *-commutative 53.1%

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

   11. Simplified 53.1%

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

Alternative 14: 37.1% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8) re (* (* 0.5 re) (+ (* im (+ 1.0 (* 0.5 im))) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * ((im * (1.0 + (0.5 * 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 <= 7.8d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * ((im * (1.0d0 + (0.5d0 * im))) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * ((im * (1.0 + (0.5 * im))) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = re
	else:
		tmp = (0.5 * re) * ((im * (1.0 + (0.5 * im))) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8)
		tmp = re;
	else
		tmp = (0.5 * re) * ((im * (1.0 + (0.5 * im))) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], re, N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 62.4%

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

   6. Taylor expanded in im around 0 40.9%

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

   if 7.79999999999999982 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in im around 0 37.9%

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

      1. distribute-lft-in 37.9%

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

      2. *-commutative 37.9%

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

      3. distribute-lft-in 37.9%

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

   10. Simplified 37.9%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.3% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8200:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 - 0.5 \cdot im\right) - 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8200.0) re (* re (- (* im (- -1.0 (* 0.5 im))) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8200.0) {
		tmp = re;
	} else {
		tmp = re * ((im * (-1.0 - (0.5 * 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 (re <= 8200.0d0) then
        tmp = re
    else
        tmp = re * ((im * ((-1.0d0) - (0.5d0 * im))) - 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8200.0) {
		tmp = re;
	} else {
		tmp = re * ((im * (-1.0 - (0.5 * im))) - 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8200.0:
		tmp = re
	else:
		tmp = re * ((im * (-1.0 - (0.5 * im))) - 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8200.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(Float64(im * Float64(-1.0 - Float64(0.5 * im))) - 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8200.0)
		tmp = re;
	else
		tmp = re * ((im * (-1.0 - (0.5 * im))) - 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8200.0], re, N[(re * N[(N[(im * N[(-1.0 - N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8200

   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 77.1%

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

   6. Taylor expanded in im around 0 41.3%

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

   if 8200 < 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 31.6%

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

   7. Applied egg-rr 17.4%

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

   9. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(3 + e^{im}\right) \]
   10. Step-by-step derivation

       1. fma-undefine 18.6%

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

       2. distribute-lft1-in 18.6%

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

       3. metadata-eval 18.6%

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

       4. neg-mul-1 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in im around 0 23.0%

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

       1. distribute-lft-in 28.8%

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

       2. *-commutative 28.8%

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

       3. distribute-lft-in 28.8%

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

   14. Simplified 23.0%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8200:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 - 0.5 \cdot im\right) - 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.4% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8) re (* (* 0.5 re) (+ im 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8) {
		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 <= 7.8d0) 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 <= 7.8) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8:
		tmp = re
	else:
		tmp = (0.5 * re) * (im + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8)
		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 <= 7.8)
		tmp = re;
	else
		tmp = (0.5 * re) * (im + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8], re, N[(N[(0.5 * re), $MachinePrecision] * N[(im + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 62.4%

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

   6. Taylor expanded in im around 0 40.9%

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

   if 7.79999999999999982 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 74.6%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in im around 0 13.6%

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

      1. +-commutative 13.6%

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

   10. Simplified 13.6%

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

Alternative 17: 29.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8200:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8200.0) re (* re (- (- im) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8200.0) {
		tmp = re;
	} else {
		tmp = 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 (re <= 8200.0d0) then
        tmp = re
    else
        tmp = re * (-im - 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8200.0) {
		tmp = re;
	} else {
		tmp = re * (-im - 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8200.0:
		tmp = re
	else:
		tmp = re * (-im - 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8200.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(Float64(-im) - 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8200.0)
		tmp = re;
	else
		tmp = re * (-im - 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8200.0], re, N[(re * N[((-im) - 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8200

   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 77.1%

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

   6. Taylor expanded in im around 0 41.3%

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

   if 8200 < 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 31.6%

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

   7. Applied egg-rr 17.4%

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

   9. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(3 + e^{im}\right) \]
   10. Step-by-step derivation

       1. fma-undefine 18.6%

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

       2. distribute-lft1-in 18.6%

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

       3. metadata-eval 18.6%

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

       4. neg-mul-1 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in im around 0 23.3%

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

       1. +-commutative 18.6%

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

   14. Simplified 23.3%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8200:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.0% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in re around 0 65.4%

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

6. Taylor expanded in im around 0 31.4%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))