math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 14

Speedup: 1.0×

• 49.9% 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

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

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(0.5 * Float64(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[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. sin-neg 100.0%

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

   7. *-commutative 100.0%

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

   8. *-commutative 100.0%

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

   9. sin-neg 100.0%

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

   10. cancel-sign-sub 100.0%

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

   11. +-commutative 100.0%

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

   12. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00058:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{-im} + e^{im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00058)
   (* 0.5 (* (sin re) 2.0))
   (* 0.5 (* re (+ (exp (- im)) (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00058) {
		tmp = 0.5 * (sin(re) * 2.0);
	} else {
		tmp = 0.5 * (re * (exp(-im) + exp(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 <= 0.00058d0) then
        tmp = 0.5d0 * (sin(re) * 2.0d0)
    else
        tmp = 0.5d0 * (re * (exp(-im) + exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00058) {
		tmp = 0.5 * (Math.sin(re) * 2.0);
	} else {
		tmp = 0.5 * (re * (Math.exp(-im) + Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00058:
		tmp = 0.5 * (math.sin(re) * 2.0)
	else:
		tmp = 0.5 * (re * (math.exp(-im) + math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00058)
		tmp = Float64(0.5 * Float64(sin(re) * 2.0));
	else
		tmp = Float64(0.5 * Float64(re * Float64(exp(Float64(-im)) + exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00058)
		tmp = 0.5 * (sin(re) * 2.0);
	else
		tmp = 0.5 * (re * (exp(-im) + exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00058], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.8e-4

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.7%

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

   6. Simplified 69.7%

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

   if 5.8e-4 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00058:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{-im} + e^{im}\right)\right)\\ \end{array} \]

Alternative 3: 65.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0)
   (* 0.5 (* (sin re) 2.0))
   (if (<= im 8.5e+68)
     (* 0.5 (pow re -512.0))
     (* 0.5 (+ (* 0.08333333333333333 (* re (pow im 4.0))) (* re 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = 0.5 * (sin(re) * 2.0);
	} else if (im <= 8.5e+68) {
		tmp = 0.5 * pow(re, -512.0);
	} else {
		tmp = 0.5 * ((0.08333333333333333 * (re * pow(im, 4.0))) + (re * 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 (im <= 720.0d0) then
        tmp = 0.5d0 * (sin(re) * 2.0d0)
    else if (im <= 8.5d+68) then
        tmp = 0.5d0 * (re ** (-512.0d0))
    else
        tmp = 0.5d0 * ((0.08333333333333333d0 * (re * (im ** 4.0d0))) + (re * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = 0.5 * (Math.sin(re) * 2.0);
	} else if (im <= 8.5e+68) {
		tmp = 0.5 * Math.pow(re, -512.0);
	} else {
		tmp = 0.5 * ((0.08333333333333333 * (re * Math.pow(im, 4.0))) + (re * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = 0.5 * (math.sin(re) * 2.0)
	elif im <= 8.5e+68:
		tmp = 0.5 * math.pow(re, -512.0)
	else:
		tmp = 0.5 * ((0.08333333333333333 * (re * math.pow(im, 4.0))) + (re * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = Float64(0.5 * Float64(sin(re) * 2.0));
	elseif (im <= 8.5e+68)
		tmp = Float64(0.5 * (re ^ -512.0));
	else
		tmp = Float64(0.5 * Float64(Float64(0.08333333333333333 * Float64(re * (im ^ 4.0))) + Float64(re * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = 0.5 * (sin(re) * 2.0);
	elseif (im <= 8.5e+68)
		tmp = 0.5 * (re ^ -512.0);
	else
		tmp = 0.5 * ((0.08333333333333333 * (re * (im ^ 4.0))) + (re * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+68], N[(0.5 * N[Power[re, -512.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(0.08333333333333333 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 720

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.5%

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

   6. Simplified 69.5%

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

   if 720 < im < 8.49999999999999966e68

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.0%

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

   6. Applied egg-rr 56.7%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{-512}} \]

   if 8.49999999999999966e68 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.9%

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

   5. Taylor expanded in im around 0 72.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + 2 \cdot re\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 72.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + 2 \cdot re\right) + re \cdot {im}^{2}\right)} \]

      2. associate-+l+ 72.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \left(2 \cdot re + re \cdot {im}^{2}\right)\right)} \]

      3. *-commutative 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \left(\color{blue}{re \cdot 2} + re \cdot {im}^{2}\right)\right) \]

      4. distribute-lft-in 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \color{blue}{re \cdot \left(2 + {im}^{2}\right)}\right) \]

      5. +-commutative 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \color{blue}{\left({im}^{2} + 2\right)}\right) \]

      6. unpow2 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \left(\color{blue}{im \cdot im} + 2\right)\right) \]

      7. fma-def 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \]

   7. Simplified 72.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]

   8. Taylor expanded in im around 0 72.5%

      \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \color{blue}{2 \cdot re}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot 2\right)\\ \end{array} \]

Alternative 4: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 920:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 920.0)
   (* 0.5 (* (sin re) 2.0))
   (if (<= im 2.9e+69)
     (* 0.5 (pow re -512.0))
     (* 0.5 (* re (* 0.08333333333333333 (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 920.0) {
		tmp = 0.5 * (sin(re) * 2.0);
	} else if (im <= 2.9e+69) {
		tmp = 0.5 * pow(re, -512.0);
	} else {
		tmp = 0.5 * (re * (0.08333333333333333 * pow(im, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 920.0d0) then
        tmp = 0.5d0 * (sin(re) * 2.0d0)
    else if (im <= 2.9d+69) then
        tmp = 0.5d0 * (re ** (-512.0d0))
    else
        tmp = 0.5d0 * (re * (0.08333333333333333d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 920.0) {
		tmp = 0.5 * (Math.sin(re) * 2.0);
	} else if (im <= 2.9e+69) {
		tmp = 0.5 * Math.pow(re, -512.0);
	} else {
		tmp = 0.5 * (re * (0.08333333333333333 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 920.0:
		tmp = 0.5 * (math.sin(re) * 2.0)
	elif im <= 2.9e+69:
		tmp = 0.5 * math.pow(re, -512.0)
	else:
		tmp = 0.5 * (re * (0.08333333333333333 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 920.0)
		tmp = Float64(0.5 * Float64(sin(re) * 2.0));
	elseif (im <= 2.9e+69)
		tmp = Float64(0.5 * (re ^ -512.0));
	else
		tmp = Float64(0.5 * Float64(re * Float64(0.08333333333333333 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 920.0)
		tmp = 0.5 * (sin(re) * 2.0);
	elseif (im <= 2.9e+69)
		tmp = 0.5 * (re ^ -512.0);
	else
		tmp = 0.5 * (re * (0.08333333333333333 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 920.0], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.9e+69], N[(0.5 * N[Power[re, -512.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 920

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.5%

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

   6. Simplified 69.5%

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

   if 920 < im < 2.8999999999999998e69

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.0%

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

   6. Applied egg-rr 56.7%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{-512}} \]

   if 2.8999999999999998e69 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.9%

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

   5. Taylor expanded in im around 0 72.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + 2 \cdot re\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 72.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + 2 \cdot re\right) + re \cdot {im}^{2}\right)} \]

      2. associate-+l+ 72.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \left(2 \cdot re + re \cdot {im}^{2}\right)\right)} \]

      3. *-commutative 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \left(\color{blue}{re \cdot 2} + re \cdot {im}^{2}\right)\right) \]

      4. distribute-lft-in 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + \color{blue}{re \cdot \left(2 + {im}^{2}\right)}\right) \]

      5. +-commutative 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \color{blue}{\left({im}^{2} + 2\right)}\right) \]

      6. unpow2 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \left(\color{blue}{im \cdot im} + 2\right)\right) \]

      7. fma-def 72.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \]

   7. Simplified 72.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right) + re \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]

   8. Taylor expanded in im around inf 72.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 72.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(0.08333333333333333 \cdot re\right) \cdot {im}^{4}\right)} \]

      2. *-commutative 72.5%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot 0.08333333333333333\right)} \cdot {im}^{4}\right) \]

      3. associate-*r* 72.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   10. Simplified 72.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 920:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right)\\ \end{array} \]

Alternative 5: 48.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 800 \lor \neg \left(im \leq 4.8 \cdot 10^{+95}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 800.0) (not (<= im 4.8e+95)))
   (* 0.5 (+ (* re 2.0) (* re (* im im))))
   (* 0.5 (pow re -512.0))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 800.0) || !(im <= 4.8e+95)) {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	} else {
		tmp = 0.5 * pow(re, -512.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 800.0d0) .or. (.not. (im <= 4.8d+95))) then
        tmp = 0.5d0 * ((re * 2.0d0) + (re * (im * im)))
    else
        tmp = 0.5d0 * (re ** (-512.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 800.0) || !(im <= 4.8e+95)) {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	} else {
		tmp = 0.5 * Math.pow(re, -512.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 800.0) or not (im <= 4.8e+95):
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)))
	else:
		tmp = 0.5 * math.pow(re, -512.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 800.0) || !(im <= 4.8e+95))
		tmp = Float64(0.5 * Float64(Float64(re * 2.0) + Float64(re * Float64(im * im))));
	else
		tmp = Float64(0.5 * (re ^ -512.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 800.0) || ~((im <= 4.8e+95)))
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	else
		tmp = 0.5 * (re ^ -512.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 800.0], N[Not[LessEqual[im, 4.8e+95]], $MachinePrecision]], N[(0.5 * N[(N[(re * 2.0), $MachinePrecision] + N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[re, -512.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 800 or 4.8000000000000001e95 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 59.5%

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

   5. Taylor expanded in im around 0 48.8%

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

      1. *-commutative 48.8%

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

      2. distribute-rgt-in 48.8%

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

      3. unpow2 48.8%

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

      4. fma-def 48.8%

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

   7. Simplified 48.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]
   8. Step-by-step derivation

      1. fma-udef 48.8%

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

      2. distribute-rgt-in 48.8%

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

   9. Applied egg-rr 48.8%

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

   if 800 < im < 4.8000000000000001e95

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 83.3%

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

   6. Applied egg-rr 50.4%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{-512}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 800 \lor \neg \left(im \leq 4.8 \cdot 10^{+95}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \end{array} \]

Alternative 6: 62.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 950:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 950.0)
   (* 0.5 (* (sin re) 2.0))
   (if (<= im 1.95e+95)
     (* 0.5 (pow re -512.0))
     (* 0.5 (+ (* re 2.0) (* re (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 950.0) {
		tmp = 0.5 * (sin(re) * 2.0);
	} else if (im <= 1.95e+95) {
		tmp = 0.5 * pow(re, -512.0);
	} else {
		tmp = 0.5 * ((re * 2.0) + (re * (im * 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 <= 950.0d0) then
        tmp = 0.5d0 * (sin(re) * 2.0d0)
    else if (im <= 1.95d+95) then
        tmp = 0.5d0 * (re ** (-512.0d0))
    else
        tmp = 0.5d0 * ((re * 2.0d0) + (re * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 950.0) {
		tmp = 0.5 * (Math.sin(re) * 2.0);
	} else if (im <= 1.95e+95) {
		tmp = 0.5 * Math.pow(re, -512.0);
	} else {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 950.0:
		tmp = 0.5 * (math.sin(re) * 2.0)
	elif im <= 1.95e+95:
		tmp = 0.5 * math.pow(re, -512.0)
	else:
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 950.0)
		tmp = Float64(0.5 * Float64(sin(re) * 2.0));
	elseif (im <= 1.95e+95)
		tmp = Float64(0.5 * (re ^ -512.0));
	else
		tmp = Float64(0.5 * Float64(Float64(re * 2.0) + Float64(re * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 950.0)
		tmp = 0.5 * (sin(re) * 2.0);
	elseif (im <= 1.95e+95)
		tmp = 0.5 * (re ^ -512.0);
	else
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 950.0], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.95e+95], N[(0.5 * N[Power[re, -512.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(re * 2.0), $MachinePrecision] + N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 950

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.5%

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

   6. Simplified 69.5%

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

   if 950 < im < 1.9499999999999999e95

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 83.3%

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

   6. Applied egg-rr 50.4%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{-512}} \]

   if 1.9499999999999999e95 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 72.0%

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

   5. Taylor expanded in im around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. distribute-rgt-in 60.5%

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

      3. unpow2 60.5%

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

      4. fma-def 60.5%

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

   7. Simplified 60.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]
   8. Step-by-step derivation

      1. fma-udef 60.5%

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

      2. distribute-rgt-in 60.5%

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

   9. Applied egg-rr 60.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 950:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot 2\right)\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot {re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 48.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+19} \lor \neg \left(im \leq 1.55 \cdot 10^{+149}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 5.6e+19) (not (<= im 1.55e+149)))
   (* 0.5 (+ (* re 2.0) (* re (* im im))))
   (*
    0.5
    (+
     0.08333333333333333
     (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 5.6e+19) || !(im <= 1.55e+149)) {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	} else {
		tmp = 0.5 * (0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 5.6d+19) .or. (.not. (im <= 1.55d+149))) then
        tmp = 0.5d0 * ((re * 2.0d0) + (re * (im * im)))
    else
        tmp = 0.5d0 * (0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 5.6e+19) || !(im <= 1.55e+149)) {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	} else {
		tmp = 0.5 * (0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 5.6e+19) or not (im <= 1.55e+149):
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)))
	else:
		tmp = 0.5 * (0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 5.6e+19) || !(im <= 1.55e+149))
		tmp = Float64(0.5 * Float64(Float64(re * 2.0) + Float64(re * Float64(im * im))));
	else
		tmp = Float64(0.5 * Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 5.6e+19) || ~((im <= 1.55e+149)))
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	else
		tmp = 0.5 * (0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 5.6e+19], N[Not[LessEqual[im, 1.55e+149]], $MachinePrecision]], N[(0.5 * N[(N[(re * 2.0), $MachinePrecision] + N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.6e19 or 1.54999999999999993e149 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 59.9%

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

   5. Taylor expanded in im around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. distribute-rgt-in 50.7%

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

      3. unpow2 50.7%

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

      4. fma-def 50.7%

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

   7. Simplified 50.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]
   8. Step-by-step derivation

      1. fma-udef 50.7%

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

      2. distribute-rgt-in 50.7%

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

   9. Applied egg-rr 50.7%

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

   if 5.6e19 < im < 1.54999999999999993e149

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 25.9%

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

   6. Taylor expanded in re around 0 36.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 36.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right)\right) \]

      2. metadata-eval 36.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)\right) \]

      3. unpow2 36.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right)\right) \]

      4. unpow2 36.5%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   8. Simplified 36.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + 0.016666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+19} \lor \neg \left(im \leq 1.55 \cdot 10^{+149}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 8: 47.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+19} \lor \neg \left(im \leq 1.15 \cdot 10^{+69}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 5.6e+19) (not (<= im 1.15e+69)))
   (* 0.5 (+ (* re 2.0) (* re (* im im))))
   (* 0.5 (+ 0.08333333333333333 (/ 0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 5.6e+19) || !(im <= 1.15e+69)) {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	} else {
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * 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 <= 5.6d+19) .or. (.not. (im <= 1.15d+69))) then
        tmp = 0.5d0 * ((re * 2.0d0) + (re * (im * im)))
    else
        tmp = 0.5d0 * (0.08333333333333333d0 + (0.25d0 / (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 5.6e+19) || !(im <= 1.15e+69)) {
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	} else {
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 5.6e+19) or not (im <= 1.15e+69):
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)))
	else:
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 5.6e+19) || !(im <= 1.15e+69))
		tmp = Float64(0.5 * Float64(Float64(re * 2.0) + Float64(re * Float64(im * im))));
	else
		tmp = Float64(0.5 * Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 5.6e+19) || ~((im <= 1.15e+69)))
		tmp = 0.5 * ((re * 2.0) + (re * (im * im)));
	else
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 5.6e+19], N[Not[LessEqual[im, 1.15e+69]], $MachinePrecision]], N[(0.5 * N[(N[(re * 2.0), $MachinePrecision] + N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.6e19 or 1.15000000000000008e69 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 60.8%

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

   5. Taylor expanded in im around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. distribute-rgt-in 47.3%

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

      3. unpow2 47.3%

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

      4. fma-def 47.3%

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

   7. Simplified 47.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]
   8. Step-by-step derivation

      1. fma-udef 47.3%

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

      2. distribute-rgt-in 47.3%

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

   9. Applied egg-rr 47.3%

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

   if 5.6e19 < im < 1.15000000000000008e69

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 44.6%

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

   6. Taylor expanded in re around 0 44.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 44.6%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      2. metadata-eval 44.6%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      3. unpow2 44.6%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 44.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+19} \lor \neg \left(im \leq 1.15 \cdot 10^{+69}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot 2 + re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)\\ \end{array} \]

Alternative 9: 34.1% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.6e+19)
   (* 0.5 (+ re re))
   (if (<= im 8.5e+69)
     (* 0.5 (+ 0.08333333333333333 (/ 0.25 (* re re))))
     (* 0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.6e+19) {
		tmp = 0.5 * (re + re);
	} else if (im <= 8.5e+69) {
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)));
	} else {
		tmp = 0.5 * (im * (re * 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 <= 5.6d+19) then
        tmp = 0.5d0 * (re + re)
    else if (im <= 8.5d+69) then
        tmp = 0.5d0 * (0.08333333333333333d0 + (0.25d0 / (re * re)))
    else
        tmp = 0.5d0 * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.6e+19) {
		tmp = 0.5 * (re + re);
	} else if (im <= 8.5e+69) {
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)));
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.6e+19:
		tmp = 0.5 * (re + re)
	elif im <= 8.5e+69:
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)))
	else:
		tmp = 0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.6e+19)
		tmp = Float64(0.5 * Float64(re + re));
	elseif (im <= 8.5e+69)
		tmp = Float64(0.5 * Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.6e+19)
		tmp = 0.5 * (re + re);
	elseif (im <= 8.5e+69)
		tmp = 0.5 * (0.08333333333333333 + (0.25 / (re * re)));
	else
		tmp = 0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.6e+19], N[(0.5 * N[(re + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+69], N[(0.5 * N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.6e19

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 56.0%

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

   5. Taylor expanded in im around 0 32.7%

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

      1. count-2 32.7%

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

   7. Simplified 32.7%

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

   if 5.6e19 < im < 8.5000000000000002e69

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 44.6%

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

   6. Taylor expanded in re around 0 44.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 44.6%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      2. metadata-eval 44.6%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      3. unpow2 44.6%

         \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 44.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)} \]

   if 8.5000000000000002e69 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.9%

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

   5. Taylor expanded in im around 0 54.3%

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

      1. *-commutative 54.3%

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

      2. distribute-rgt-in 54.3%

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

      3. unpow2 54.3%

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

      4. fma-def 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in im around inf 54.3%

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

      1. unpow2 54.3%

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

      2. *-commutative 54.3%

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

      3. associate-*l* 41.4%

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

   10. Simplified 41.4%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 34.2% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4500000:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4500000.0)
   (* 0.5 (+ re re))
   (if (<= im 7.2e+68) (* 0.5 (/ 0.25 (* re re))) (* 0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4500000.0) {
		tmp = 0.5 * (re + re);
	} else if (im <= 7.2e+68) {
		tmp = 0.5 * (0.25 / (re * re));
	} else {
		tmp = 0.5 * (im * (re * 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 <= 4500000.0d0) then
        tmp = 0.5d0 * (re + re)
    else if (im <= 7.2d+68) then
        tmp = 0.5d0 * (0.25d0 / (re * re))
    else
        tmp = 0.5d0 * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4500000.0) {
		tmp = 0.5 * (re + re);
	} else if (im <= 7.2e+68) {
		tmp = 0.5 * (0.25 / (re * re));
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4500000.0:
		tmp = 0.5 * (re + re)
	elif im <= 7.2e+68:
		tmp = 0.5 * (0.25 / (re * re))
	else:
		tmp = 0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4500000.0)
		tmp = Float64(0.5 * Float64(re + re));
	elseif (im <= 7.2e+68)
		tmp = Float64(0.5 * Float64(0.25 / Float64(re * re)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4500000.0)
		tmp = 0.5 * (re + re);
	elseif (im <= 7.2e+68)
		tmp = 0.5 * (0.25 / (re * re));
	else
		tmp = 0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4500000.0], N[(0.5 * N[(re + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+68], N[(0.5 * N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5e6

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 56.3%

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

   5. Taylor expanded in im around 0 32.9%

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

      1. count-2 32.9%

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

   7. Simplified 32.9%

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

   if 4.5e6 < im < 7.1999999999999998e68

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 41.6%

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

   6. Taylor expanded in re around 0 41.6%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 41.6%

         \[\leadsto 0.5 \cdot \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 41.6%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{0.25}{re \cdot re}} \]

   if 7.1999999999999998e68 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.9%

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

   5. Taylor expanded in im around 0 54.3%

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

      1. *-commutative 54.3%

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

      2. distribute-rgt-in 54.3%

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

      3. unpow2 54.3%

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

      4. fma-def 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in im around inf 54.3%

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

      1. unpow2 54.3%

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

      2. *-commutative 54.3%

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

      3. associate-*l* 41.4%

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

   10. Simplified 41.4%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4500000:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 33.9% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) (* 0.5 (+ re re)) (* 0.5 (* im (* re im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = 0.5 * (re + re);
	} else {
		tmp = 0.5 * (im * (re * 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 <= 1.4d0) then
        tmp = 0.5d0 * (re + re)
    else
        tmp = 0.5d0 * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = 0.5 * (re + re);
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = 0.5 * (re + re)
	else:
		tmp = 0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = Float64(0.5 * Float64(re + re));
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = 0.5 * (re + re);
	else
		tmp = 0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], N[(0.5 * N[(re + re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 56.1%

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

   5. Taylor expanded in im around 0 33.0%

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

      1. count-2 33.0%

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

   7. Simplified 33.0%

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

   if 1.3999999999999999 < 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. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. sin-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. sin-neg 100.0%

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

      10. cancel-sign-sub 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.7%

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

   5. Taylor expanded in im around 0 44.3%

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

      1. *-commutative 44.3%

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

      2. distribute-rgt-in 44.3%

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

      3. unpow2 44.3%

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

      4. fma-def 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in im around inf 44.3%

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

      1. unpow2 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*l* 34.2%

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

   10. Simplified 34.2%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 12: 26.4% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(re + re), $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. associate-*l* 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. sin-neg 100.0%

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

   7. *-commutative 100.0%

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

   8. *-commutative 100.0%

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

   9. sin-neg 100.0%

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

   10. cancel-sign-sub 100.0%

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

   11. +-commutative 100.0%

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

   12. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 61.8%

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

5. Taylor expanded in im around 0 24.2%

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

   1. count-2 24.2%

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

7. Simplified 24.2%

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

8. Final simplification 24.2%

   \[\leadsto 0.5 \cdot \left(re + re\right) \]

Alternative 13: 6.8% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * re), $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. associate-*l* 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. sin-neg 100.0%

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

   7. *-commutative 100.0%

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

   8. *-commutative 100.0%

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

   9. sin-neg 100.0%

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

   10. cancel-sign-sub 100.0%

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

   11. +-commutative 100.0%

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

   12. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 61.8%

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

6. Applied egg-rr 4.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left|re\right|} \]
7. Step-by-step derivation

   1. unpow1 4.5%

      \[\leadsto 0.5 \cdot \left|\color{blue}{{re}^{1}}\right| \]

   2. sqr-pow 3.1%

      \[\leadsto 0.5 \cdot \left|\color{blue}{{re}^{\left(\frac{1}{2}\right)} \cdot {re}^{\left(\frac{1}{2}\right)}}\right| \]

   3. fabs-sqr 3.1%

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

   4. sqr-pow 6.4%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{1}} \]

   5. unpow1 6.4%

      \[\leadsto 0.5 \cdot \color{blue}{re} \]

8. Simplified 6.4%

   \[\leadsto 0.5 \cdot \color{blue}{re} \]

9. Final simplification 6.4%

   \[\leadsto 0.5 \cdot re \]

Alternative 14: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.041666666666666664 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.041666666666666664)

C

double code(double re, double im) {
	return 0.041666666666666664;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.041666666666666664d0
end function

Java

public static double code(double re, double im) {
	return 0.041666666666666664;
}

Python

def code(re, im):
	return 0.041666666666666664

Julia

function code(re, im)
	return 0.041666666666666664
end

MATLAB

function tmp = code(re, im)
	tmp = 0.041666666666666664;
end

Wolfram

code[re_, im_] := 0.041666666666666664

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. associate-*l* 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. sin-neg 100.0%

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

   7. *-commutative 100.0%

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

   8. *-commutative 100.0%

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

   9. sin-neg 100.0%

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

   10. cancel-sign-sub 100.0%

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

   11. +-commutative 100.0%

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

   12. *-commutative 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 12.4%

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

6. Taylor expanded in re around 0 12.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 12.2%

      \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

   2. metadata-eval 12.2%

      \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

   3. unpow2 12.2%

      \[\leadsto 0.5 \cdot \left(0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

8. Simplified 12.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 + \frac{0.25}{re \cdot re}\right)} \]

9. Taylor expanded in re around inf 4.2%

   \[\leadsto 0.5 \cdot \color{blue}{0.08333333333333333} \]

10. Final simplification 4.2%

    \[\leadsto 0.041666666666666664 \]

Reproduce

herbie shell --seed 2023264 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))