math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 17

Speedup: 1.0×

• 48.8% 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: 74.6% 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 78.2%

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

6. Final simplification 78.2%

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

Alternative 3: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.2e-13)
   (sin re)
   (if (<= im 1e+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 <= 3.2e-13) {
		tmp = sin(re);
	} else if (im <= 1e+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 <= 3.2d-13) then
        tmp = sin(re)
    else if (im <= 1d+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 <= 3.2e-13) {
		tmp = Math.sin(re);
	} else if (im <= 1e+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 <= 3.2e-13:
		tmp = math.sin(re)
	elif im <= 1e+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 <= 3.2e-13)
		tmp = sin(re);
	elseif (im <= 1e+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 <= 3.2e-13)
		tmp = sin(re);
	elseif (im <= 1e+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, 3.2e-13], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1e+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 < 3.2e-13

   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 70.9%

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

   if 3.2e-13 < im < 1e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 68.1%

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

   6. Taylor expanded in im around 0 63.9%

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

   if 1e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   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 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{-13}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.8)
   (sin re)
   (if (<= im 1e+103)
     (+ (* 0.5 re) (* (exp im) (* 0.5 re)))
     (*
      (* 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 <= 4.8) {
		tmp = sin(re);
	} else if (im <= 1e+103) {
		tmp = (0.5 * re) + (exp(im) * (0.5 * re));
	} 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 <= 4.8d0) then
        tmp = sin(re)
    else if (im <= 1d+103) then
        tmp = (0.5d0 * re) + (exp(im) * (0.5d0 * re))
    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 <= 4.8) {
		tmp = Math.sin(re);
	} else if (im <= 1e+103) {
		tmp = (0.5 * re) + (Math.exp(im) * (0.5 * re));
	} 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 <= 4.8:
		tmp = math.sin(re)
	elif im <= 1e+103:
		tmp = (0.5 * re) + (math.exp(im) * (0.5 * re))
	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 <= 4.8)
		tmp = sin(re);
	elseif (im <= 1e+103)
		tmp = Float64(Float64(0.5 * re) + Float64(exp(im) * Float64(0.5 * re)));
	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 <= 4.8)
		tmp = sin(re);
	elseif (im <= 1e+103)
		tmp = (0.5 * re) + (exp(im) * (0.5 * re));
	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, 4.8], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1e+103], N[(N[(0.5 * re), $MachinePrecision] + N[(N[Exp[im], $MachinePrecision] * N[(0.5 * re), $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 < 4.79999999999999982

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.9%

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

   if 4.79999999999999982 < im < 1e103

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 98.1%

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

   6. Taylor expanded in re around 0 62.9%

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

      1. +-commutative 62.9%

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

      2. distribute-lft-in 62.9%

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

      3. *-rgt-identity 62.9%

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

   8. Applied egg-rr 62.9%

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

   if 1e103 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   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 74.7%

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

Alternative 5: 72.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot re + e^{im} \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.4)
   (sin re)
   (if (<= im 7e+153)
     (+ (* 0.5 re) (* (exp im) (* 0.5 re)))
     (* (* 0.5 (sin re)) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.4) {
		tmp = sin(re);
	} else if (im <= 7e+153) {
		tmp = (0.5 * re) + (exp(im) * (0.5 * re));
	} else {
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.4d0) then
        tmp = sin(re)
    else if (im <= 7d+153) then
        tmp = (0.5d0 * re) + (exp(im) * (0.5d0 * re))
    else
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.4) {
		tmp = Math.sin(re);
	} else if (im <= 7e+153) {
		tmp = (0.5 * re) + (Math.exp(im) * (0.5 * re));
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.4:
		tmp = math.sin(re)
	elif im <= 7e+153:
		tmp = (0.5 * re) + (math.exp(im) * (0.5 * re))
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.4)
		tmp = sin(re);
	elseif (im <= 7e+153)
		tmp = Float64(Float64(0.5 * re) + Float64(exp(im) * Float64(0.5 * re)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.4)
		tmp = sin(re);
	elseif (im <= 7e+153)
		tmp = (0.5 * re) + (exp(im) * (0.5 * re));
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.4], N[Sin[re], $MachinePrecision], If[LessEqual[im, 7e+153], N[(N[(0.5 * re), $MachinePrecision] + N[(N[Exp[im], $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $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 < 3.39999999999999991

   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 70.9%

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

   if 3.39999999999999991 < im < 6.9999999999999998e153

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.7%

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

   6. Taylor expanded in re around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. distribute-lft-in 68.2%

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

      3. *-rgt-identity 68.2%

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

   8. Applied egg-rr 68.2%

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

   if 6.9999999999999998e153 < 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 97.2%

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

      1. *-commutative 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot re + e^{im} \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.45:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot re + e^{im} \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.45) (sin re) (+ (* 0.5 re) (* (exp im) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.45) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) + (exp(im) * (0.5 * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.45d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) + (exp(im) * (0.5d0 * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.45) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) + (Math.exp(im) * (0.5 * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.45:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) + (math.exp(im) * (0.5 * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.45)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) + Float64(exp(im) * Float64(0.5 * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.45)
		tmp = sin(re);
	else
		tmp = (0.5 * re) + (exp(im) * (0.5 * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.45], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] + N[(N[Exp[im], $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.4500000000000002

   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 70.9%

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

   if 3.4500000000000002 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.2%

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

   6. Taylor expanded in re around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. distribute-lft-in 73.3%

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

      3. *-rgt-identity 73.3%

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

   8. Applied egg-rr 73.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.45:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot re + e^{im} \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.1) (sin re) (* (+ (exp im) 1.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.1) {
		tmp = sin(re);
	} else {
		tmp = (exp(im) + 1.0) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.1d0) then
        tmp = sin(re)
    else
        tmp = (exp(im) + 1.0d0) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.1) {
		tmp = Math.sin(re);
	} else {
		tmp = (Math.exp(im) + 1.0) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.1:
		tmp = math.sin(re)
	else:
		tmp = (math.exp(im) + 1.0) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.1)
		tmp = sin(re);
	else
		tmp = Float64(Float64(exp(im) + 1.0) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.1)
		tmp = sin(re);
	else
		tmp = (exp(im) + 1.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.1], N[Sin[re], $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.0999999999999996

   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 70.9%

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

   if 4.0999999999999996 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.2%

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

   6. Taylor expanded in re around 0 73.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 1\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{expm1}\left(im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3) (sin re) (* (* 0.5 re) (expm1 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * expm1(im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * Math.expm1(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.3:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * math.expm1(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.3)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * expm1(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.3], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(Exp[im] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2999999999999998

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.9%

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

   if 3.2999999999999998 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.2%

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

   6. Taylor expanded in re around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. distribute-lft-in 73.3%

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

      3. *-rgt-identity 73.3%

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

   8. Applied egg-rr 73.3%

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

      1. *-commutative 73.3%

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

      2. add-sqr-sqrt 35.0%

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

      3. sqrt-unprod 70.4%

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

      4. sqr-neg 70.4%

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

      5. sqrt-unprod 38.4%

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

      6. add-sqr-sqrt 73.3%

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

      7. cancel-sign-sub-inv 73.3%

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

      8. *-commutative 73.3%

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

      9. associate-*l* 73.3%

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

   10. Applied egg-rr 73.3%

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

       1. distribute-lft-out-- 73.3%

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

       2. *-rgt-identity 73.3%

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

       3. distribute-lft-out-- 73.3%

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

       4. expm1-undefine 73.3%

          \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\mathsf{expm1}\left(im\right)}\right) \]

       5. associate-*l* 73.3%

          \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \mathsf{expm1}\left(im\right)} \]

   12. Simplified 73.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \mathsf{expm1}\left(im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 11.5:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{expm1}\left(im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 11.5) (sin re) (* re (expm1 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 11.5) {
		tmp = sin(re);
	} else {
		tmp = re * expm1(im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 11.5) {
		tmp = Math.sin(re);
	} else {
		tmp = re * Math.expm1(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 11.5:
		tmp = math.sin(re)
	else:
		tmp = re * math.expm1(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 11.5)
		tmp = sin(re);
	else
		tmp = Float64(re * expm1(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 11.5], N[Sin[re], $MachinePrecision], N[(re * N[(Exp[im] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 11.5

   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 70.9%

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

   if 11.5 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.2%

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

   6. Taylor expanded in re around 0 73.3%

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

   8. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(1 + e^{im}\right) \]
   9. Step-by-step derivation

      1. fma-undefine 24.7%

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

      2. distribute-lft1-in 24.7%

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

      3. metadata-eval 24.7%

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

      4. neg-mul-1 24.7%

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

   10. Simplified 24.7%

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

       1. distribute-rgt-in 24.7%

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

       2. *-un-lft-identity 24.7%

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

       3. neg-sub0 24.7%

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

       4. associate-+l- 24.7%

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

       5. *-commutative 24.7%

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

       6. add-sqr-sqrt 10.2%

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

       7. sqrt-unprod 30.6%

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

       8. sqr-neg 30.6%

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

       9. sqrt-unprod 35.0%

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

       10. add-sqr-sqrt 72.9%

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

   12. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{0 - \left(re - re \cdot e^{im}\right)} \]
   13. Step-by-step derivation

       1. associate--r- 72.9%

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

       2. sub0-neg 72.9%

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

       3. +-commutative 72.9%

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

       4. sub-neg 72.9%

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

       5. *-rgt-identity 72.9%

          \[\leadsto re \cdot e^{im} - \color{blue}{re \cdot 1} \]

       6. distribute-lft-out-- 72.9%

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

       7. expm1-undefine 72.9%

          \[\leadsto re \cdot \color{blue}{\mathsf{expm1}\left(im\right)} \]

   14. Simplified 72.9%

       \[\leadsto \color{blue}{re \cdot \mathsf{expm1}\left(im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 13.5:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 13.5)
   (sin re)
   (*
    0.5
    (* re (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 13.5) {
		tmp = sin(re);
	} else {
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 13.5d0) then
        tmp = sin(re)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 13.5) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 13.5:
		tmp = math.sin(re)
	else:
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 13.5)
		tmp = sin(re);
	else
		tmp = Float64(0.5 * Float64(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 <= 13.5)
		tmp = sin(re);
	else
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 13.5], N[Sin[re], $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 13.5

   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 70.9%

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

   if 13.5 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.2%

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

   6. Taylor expanded in im around 0 61.4%

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

      1. *-commutative 61.4%

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

   8. Simplified 61.4%

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

   9. Taylor expanded in re around 0 54.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 13.5:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\\ \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot \left(1 + t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 - t\_0\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 0.5 (* im 0.16666666666666666)))))
   (if (<= re 4e+273)
     (* 0.5 (* re (+ 2.0 (* im (+ 1.0 t_0)))))
     (* re (- (* im (- -1.0 t_0)) 2.0)))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 + (im * 0.16666666666666666));
	double tmp;
	if (re <= 4e+273) {
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + t_0))));
	} else {
		tmp = re * ((im * (-1.0 - t_0)) - 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 + (im * 0.16666666666666666d0))
    if (re <= 4d+273) then
        tmp = 0.5d0 * (re * (2.0d0 + (im * (1.0d0 + t_0))))
    else
        tmp = re * ((im * ((-1.0d0) - t_0)) - 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 + (im * 0.16666666666666666));
	double tmp;
	if (re <= 4e+273) {
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + t_0))));
	} else {
		tmp = re * ((im * (-1.0 - t_0)) - 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 + (im * 0.16666666666666666))
	tmp = 0
	if re <= 4e+273:
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + t_0))))
	else:
		tmp = re * ((im * (-1.0 - t_0)) - 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))
	tmp = 0.0
	if (re <= 4e+273)
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * Float64(1.0 + t_0)))));
	else
		tmp = Float64(re * Float64(Float64(im * Float64(-1.0 - t_0)) - 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 + (im * 0.16666666666666666));
	tmp = 0.0;
	if (re <= 4e+273)
		tmp = 0.5 * (re * (2.0 + (im * (1.0 + t_0))));
	else
		tmp = re * ((im * (-1.0 - t_0)) - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 4e+273], N[(0.5 * N[(re * N[(2.0 + N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < 3.99999999999999978e273

   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 77.9%

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

   6. Taylor expanded in im around 0 66.9%

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

      1. *-commutative 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in re around 0 47.0%

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

   if 3.99999999999999978e273 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 1.0%

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

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(1 + e^{im}\right) \]
   9. Step-by-step derivation

      1. fma-undefine 51.0%

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

      2. distribute-lft1-in 51.0%

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

      3. metadata-eval 51.0%

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

      4. neg-mul-1 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in im around 0 51.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 51.0%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4e+273)
   (* re (+ 1.0 (* im (+ 0.5 (* im 0.25)))))
   (* re (- (* im (- -1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4e+273) {
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	} else {
		tmp = re * ((im * (-1.0 - (im * (0.5 + (im * 0.16666666666666666))))) - 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4d+273) then
        tmp = re * (1.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    else
        tmp = re * ((im * ((-1.0d0) - (im * (0.5d0 + (im * 0.16666666666666666d0))))) - 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4e+273) {
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	} else {
		tmp = re * ((im * (-1.0 - (im * (0.5 + (im * 0.16666666666666666))))) - 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4e+273:
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))))
	else:
		tmp = re * ((im * (-1.0 - (im * (0.5 + (im * 0.16666666666666666))))) - 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4e+273)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	else
		tmp = Float64(re * Float64(Float64(im * Float64(-1.0 - Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) - 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4e+273)
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	else
		tmp = re * ((im * (-1.0 - (im * (0.5 + (im * 0.16666666666666666))))) - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4e+273], N[(re * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * N[(-1.0 - N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.99999999999999978e273

   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 77.9%

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

   6. Taylor expanded in re around 0 49.0%

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

   7. Taylor expanded in im around 0 45.0%

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

   8. Taylor expanded in re around 0 49.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]

   if 3.99999999999999978e273 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 1.0%

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

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(1 + e^{im}\right) \]
   9. Step-by-step derivation

      1. fma-undefine 51.0%

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

      2. distribute-lft1-in 51.0%

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

      3. metadata-eval 51.0%

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

      4. neg-mul-1 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in im around 0 51.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 51.0%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.3% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4e+273)
   (* re (+ 1.0 (* im (+ 0.5 (* im 0.25)))))
   (* re (- (- im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4e+273) {
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	} else {
		tmp = re * (-im - 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4d+273) then
        tmp = re * (1.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    else
        tmp = re * (-im - 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4e+273) {
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	} else {
		tmp = re * (-im - 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4e+273:
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))))
	else:
		tmp = re * (-im - 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4e+273)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	else
		tmp = Float64(re * Float64(Float64(-im) - 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4e+273)
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	else
		tmp = re * (-im - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4e+273], N[(re * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[((-im) - 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.99999999999999978e273

   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 77.9%

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

   6. Taylor expanded in re around 0 49.0%

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

   7. Taylor expanded in im around 0 45.0%

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

   8. Taylor expanded in re around 0 49.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]

   if 3.99999999999999978e273 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 1.0%

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

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(1 + e^{im}\right) \]
   9. Step-by-step derivation

      1. fma-undefine 51.0%

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

      2. distribute-lft1-in 51.0%

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

      3. metadata-eval 51.0%

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

      4. neg-mul-1 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in im around 0 51.0%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.5% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4e+273) (+ re (* 0.5 (* re im))) (* re (- (- im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4e+273) {
		tmp = re + (0.5 * (re * im));
	} else {
		tmp = re * (-im - 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4d+273) then
        tmp = re + (0.5d0 * (re * im))
    else
        tmp = re * (-im - 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4e+273) {
		tmp = re + (0.5 * (re * im));
	} else {
		tmp = re * (-im - 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4e+273:
		tmp = re + (0.5 * (re * im))
	else:
		tmp = re * (-im - 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4e+273)
		tmp = Float64(re + Float64(0.5 * Float64(re * im)));
	else
		tmp = Float64(re * Float64(Float64(-im) - 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4e+273)
		tmp = re + (0.5 * (re * im));
	else
		tmp = re * (-im - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4e+273], N[(re + N[(0.5 * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[((-im) - 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.99999999999999978e273

   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 77.9%

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

   6. Taylor expanded in re around 0 49.0%

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

   7. Taylor expanded in im around 0 32.2%

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

   if 3.99999999999999978e273 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 1.0%

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

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(1 + e^{im}\right) \]
   9. Step-by-step derivation

      1. fma-undefine 51.0%

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

      2. distribute-lft1-in 51.0%

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

      3. metadata-eval 51.0%

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

      4. neg-mul-1 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in im around 0 51.0%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+273}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 210000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 210000.0) re (* re (- (- im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 210000.0) {
		tmp = re;
	} else {
		tmp = re * (-im - 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 210000.0d0) then
        tmp = re
    else
        tmp = re * (-im - 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 210000.0) {
		tmp = re;
	} else {
		tmp = re * (-im - 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 210000.0:
		tmp = re
	else:
		tmp = re * (-im - 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 210000.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(Float64(-im) - 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 210000.0)
		tmp = re;
	else
		tmp = re * (-im - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 210000.0], re, N[(re * N[((-im) - 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.1e5

   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 78.1%

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

   6. Taylor expanded in im around 0 38.5%

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

   if 2.1e5 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.3%

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

   6. Taylor expanded in re around 0 13.1%

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

   8. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, re, re\right)} \cdot \left(1 + e^{im}\right) \]
   9. Step-by-step derivation

      1. fma-undefine 17.7%

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

      2. distribute-lft1-in 17.7%

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

      3. metadata-eval 17.7%

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

      4. neg-mul-1 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in im around 0 19.5%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 210000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(-im\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.9% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.235:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 0.235) re 0.75))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.235) {
		tmp = re;
	} else {
		tmp = 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.235d0) then
        tmp = re
    else
        tmp = 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.235) {
		tmp = re;
	} else {
		tmp = 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.235:
		tmp = re
	else:
		tmp = 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.235)
		tmp = re;
	else
		tmp = 0.75;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.235)
		tmp = re;
	else
		tmp = 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.235], re, 0.75]

Derivation

1. Split input into 2 regimes

2. if re < 0.23499999999999999

   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 78.4%

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

   6. Taylor expanded in im around 0 38.8%

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

   if 0.23499999999999999 < 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 27.0%

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

   7. Applied egg-rr 2.8%

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

   9. Applied egg-rr 2.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - -2\right)} \cdot 0.25 \]
   10. Step-by-step derivation

       1. log1p-undefine 2.8%

          \[\leadsto \left(e^{\color{blue}{\log \left(1 + re\right)}} - -2\right) \cdot 0.25 \]

       2. rem-exp-log 2.8%

          \[\leadsto \left(\color{blue}{\left(1 + re\right)} - -2\right) \cdot 0.25 \]

       3. +-commutative 2.8%

          \[\leadsto \left(\color{blue}{\left(re + 1\right)} - -2\right) \cdot 0.25 \]

       4. associate--l+ 2.8%

          \[\leadsto \color{blue}{\left(re + \left(1 - -2\right)\right)} \cdot 0.25 \]

       5. metadata-eval 2.8%

          \[\leadsto \left(re + \color{blue}{3}\right) \cdot 0.25 \]

   11. Simplified 2.8%

       \[\leadsto \color{blue}{\left(re + 3\right)} \cdot 0.25 \]

   12. Taylor expanded in re around 0 5.6%

       \[\leadsto \color{blue}{0.75} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 4.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

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 64.9%

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

7. Applied egg-rr 6.5%

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

9. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - -2\right)} \cdot 0.25 \]
10. Step-by-step derivation

    1. log1p-undefine 2.2%

       \[\leadsto \left(e^{\color{blue}{\log \left(1 + re\right)}} - -2\right) \cdot 0.25 \]

    2. rem-exp-log 2.9%

       \[\leadsto \left(\color{blue}{\left(1 + re\right)} - -2\right) \cdot 0.25 \]

    3. +-commutative 2.9%

       \[\leadsto \left(\color{blue}{\left(re + 1\right)} - -2\right) \cdot 0.25 \]

    4. associate--l+ 2.9%

       \[\leadsto \color{blue}{\left(re + \left(1 - -2\right)\right)} \cdot 0.25 \]

    5. metadata-eval 2.9%

       \[\leadsto \left(re + \color{blue}{3}\right) \cdot 0.25 \]

11. Simplified 2.9%

    \[\leadsto \color{blue}{\left(re + 3\right)} \cdot 0.25 \]

12. Taylor expanded in re around 0 4.0%

    \[\leadsto \color{blue}{0.75} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))