math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 16

Speedup: 1.0×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01 \lor \neg \left(\sin re \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (sin re) -0.01) (not (<= (sin re) 5e-16)))
   (*
    (* 0.5 (sin re))
    (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333)))
   (* 0.5 (+ (/ re (exp im)) (* re (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if ((sin(re) <= -0.01) || !(sin(re) <= 5e-16)) {
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
	} else {
		tmp = 0.5 * ((re / exp(im)) + (re * 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 ((sin(re) <= (-0.01d0)) .or. (.not. (sin(re) <= 5d-16))) then
        tmp = (0.5d0 * sin(re)) * ((2.0d0 + (im * im)) + ((im ** 4.0d0) * 0.08333333333333333d0))
    else
        tmp = 0.5d0 * ((re / exp(im)) + (re * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sin(re) <= -0.01) || !(Math.sin(re) <= 5e-16)) {
		tmp = (0.5 * Math.sin(re)) * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
	} else {
		tmp = 0.5 * ((re / Math.exp(im)) + (re * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sin(re) <= -0.01) or not (math.sin(re) <= 5e-16):
		tmp = (0.5 * math.sin(re)) * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))
	else:
		tmp = 0.5 * ((re / math.exp(im)) + (re * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((sin(re) <= -0.01) || !(sin(re) <= 5e-16))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)));
	else
		tmp = Float64(0.5 * Float64(Float64(re / exp(im)) + Float64(re * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sin(re) <= -0.01) || ~((sin(re) <= 5e-16)))
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + ((im ^ 4.0) * 0.08333333333333333));
	else
		tmp = 0.5 * ((re / exp(im)) + (re * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[Not[LessEqual[N[Sin[re], $MachinePrecision], 5e-16]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(re / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(re * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002 or 5.0000000000000004e-16 < (sin.f64 re)

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.1%

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

      1. associate-+r+ 91.1%

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

      2. unpow2 91.1%

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

   6. Simplified 91.1%

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

   if -0.0100000000000000002 < (sin.f64 re) < 5.0000000000000004e-16

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. fma-def 100.0%

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

      3. exp-neg 100.0%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{im}, re, \color{blue}{\frac{1}{e^{im}}} \cdot re\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{im}, re, \color{blue}{\frac{1 \cdot re}{e^{im}}}\right) \]

      5. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01 \lor \neg \left(\sin re \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \end{array} \]

Alternative 3: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -9 \cdot 10^{+96}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 370:\\ \;\;\;\;t_0 \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(t_0 \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -9e+96)
     (* (pow im 4.0) (* (sin re) 0.041666666666666664))
     (if (<= im -31000000.0)
       (* (+ (exp (- im)) (exp im)) (* 0.5 re))
       (if (<= im 370.0)
         (* t_0 (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333)))
         (log1p (expm1 (* im (* t_0 im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -9e+96) {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	} else if (im <= -31000000.0) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else if (im <= 370.0) {
		tmp = t_0 * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
	} else {
		tmp = log1p(expm1((im * (t_0 * im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -9e+96) {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	} else if (im <= -31000000.0) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else if (im <= 370.0) {
		tmp = t_0 * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
	} else {
		tmp = Math.log1p(Math.expm1((im * (t_0 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -9e+96:
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	elif im <= -31000000.0:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	elif im <= 370.0:
		tmp = t_0 * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))
	else:
		tmp = math.log1p(math.expm1((im * (t_0 * im))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -9e+96)
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	elseif (im <= -31000000.0)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	elseif (im <= 370.0)
		tmp = Float64(t_0 * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)));
	else
		tmp = log1p(expm1(Float64(im * Float64(t_0 * im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+96], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -31000000.0], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 370.0], N[(t$95$0 * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(im * N[(t$95$0 * im), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -8.99999999999999914e96

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{4}\right) \cdot 0.041666666666666664} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \sin re\right)} \cdot 0.041666666666666664 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   if -8.99999999999999914e96 < im < -3.1e7

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -3.1e7 < im < 370

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.6%

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

      1. associate-+r+ 97.6%

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

      2. unpow2 97.6%

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

   6. Simplified 97.6%

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

   if 370 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 58.3%

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

      1. unpow2 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in im around inf 58.3%

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

      1. associate-*r* 58.3%

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

      2. *-commutative 58.3%

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

      3. unpow2 58.3%

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

   9. Simplified 58.3%

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

       1. log1p-expm1-u 94.9%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)\right)\right)} \]

       2. associate-*l* 94.9%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(im \cdot \left(0.5 \cdot \sin re\right)\right)}\right)\right) \]

   11. Applied egg-rr 94.9%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(im \cdot \left(0.5 \cdot \sin re\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+96}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 370:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(\left(0.5 \cdot \sin re\right) \cdot im\right)\right)\right)\\ \end{array} \]

Alternative 4: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := {im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq -9 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.2:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* (pow im 4.0) (* (sin re) 0.041666666666666664))))
   (if (<= im -9e+96)
     t_1
     (if (<= im -31000000.0)
       t_0
       (if (<= im 6.2)
         (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
         (if (<= im 1.15e+77) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double t_1 = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -9e+96) {
		tmp = t_1;
	} else if (im <= -31000000.0) {
		tmp = t_0;
	} else if (im <= 6.2) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(-im) + exp(im)) * (0.5d0 * re)
    t_1 = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    if (im <= (-9d+96)) then
        tmp = t_1
    else if (im <= (-31000000.0d0)) then
        tmp = t_0
    else if (im <= 6.2d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 1.15d+77) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double t_1 = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -9e+96) {
		tmp = t_1;
	} else if (im <= -31000000.0) {
		tmp = t_0;
	} else if (im <= 6.2) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	t_1 = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	tmp = 0
	if im <= -9e+96:
		tmp = t_1
	elif im <= -31000000.0:
		tmp = t_0
	elif im <= 6.2:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 1.15e+77:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	t_1 = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664))
	tmp = 0.0
	if (im <= -9e+96)
		tmp = t_1;
	elseif (im <= -31000000.0)
		tmp = t_0;
	elseif (im <= 6.2)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	t_1 = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	tmp = 0.0;
	if (im <= -9e+96)
		tmp = t_1;
	elseif (im <= -31000000.0)
		tmp = t_0;
	elseif (im <= 6.2)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 1.15e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+96], t$95$1, If[LessEqual[im, -31000000.0], t$95$0, If[LessEqual[im, 6.2], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.99999999999999914e96 or 1.14999999999999997e77 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{4}\right) \cdot 0.041666666666666664} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \sin re\right)} \cdot 0.041666666666666664 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   if -8.99999999999999914e96 < im < -3.1e7 or 6.20000000000000018 < im < 1.14999999999999997e77

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 87.5%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

   6. Simplified 87.5%

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

   if -3.1e7 < im < 6.20000000000000018

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.1%

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

      1. unpow2 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+96}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 6.2:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 5: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq -3.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(e^{im}, re, re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 4.0) (* (sin re) 0.041666666666666664))))
   (if (<= im -3.7)
     t_0
     (if (<= im 2.1e+21)
       (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
       (if (<= im 1.15e+77) (* 0.5 (fma (exp im) re (* re re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -3.7) {
		tmp = t_0;
	} else if (im <= 2.1e+21) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * fma(exp(im), re, (re * re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664))
	tmp = 0.0
	if (im <= -3.7)
		tmp = t_0;
	elseif (im <= 2.1e+21)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * fma(exp(im), re, Float64(re * re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.7], t$95$0, If[LessEqual[im, 2.1e+21], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[im], $MachinePrecision] * re + N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.7000000000000002 or 1.14999999999999997e77 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 89.4%

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

      1. associate-+r+ 89.4%

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

      2. unpow2 89.4%

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

   6. Simplified 89.4%

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

   7. Taylor expanded in im around inf 89.4%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 89.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{4}\right) \cdot 0.041666666666666664} \]

      2. *-commutative 89.4%

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \sin re\right)} \cdot 0.041666666666666664 \]

      3. associate-*l* 89.4%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   9. Simplified 89.4%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   if -3.7000000000000002 < im < 2.1e21

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. unpow2 97.0%

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

   6. Simplified 97.0%

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

   if 2.1e21 < im < 1.14999999999999997e77

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 71.4%

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

      1. distribute-rgt-in 71.4%

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

      2. fma-def 71.4%

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

      3. exp-neg 71.4%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{im}, re, \color{blue}{\frac{1}{e^{im}}} \cdot re\right) \]

      4. associate-*l/ 71.4%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{im}, re, \color{blue}{\frac{1 \cdot re}{e^{im}}}\right) \]

      5. *-lft-identity 71.4%

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

   6. Simplified 71.4%

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

   8. Applied egg-rr 71.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.7:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(e^{im}, re, re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 6: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.7 \lor \neg \left(im \leq 3.7\right):\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.7) (not (<= im 3.7)))
   (* (pow im 4.0) (* (sin re) 0.041666666666666664))
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.7) || !(im <= 3.7)) {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	} else {
		tmp = (0.5 * sin(re)) * (2.0 + (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 <= (-3.7d0)) .or. (.not. (im <= 3.7d0))) then
        tmp = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    else
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.7) || !(im <= 3.7)) {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	} else {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.7) or not (im <= 3.7):
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	else:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.7) || !(im <= 3.7))
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.7) || ~((im <= 3.7)))
		tmp = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	else
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.7], N[Not[LessEqual[im, 3.7]], $MachinePrecision]], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.7000000000000002 or 3.7000000000000002 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.2%

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

      1. associate-+r+ 82.2%

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

      2. unpow2 82.2%

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

   6. Simplified 82.2%

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

   7. Taylor expanded in im around inf 82.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 82.2%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{4}\right) \cdot 0.041666666666666664} \]

      2. *-commutative 82.2%

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \sin re\right)} \cdot 0.041666666666666664 \]

      3. associate-*l* 82.2%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   9. Simplified 82.2%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)} \]

   if -3.7000000000000002 < im < 3.7000000000000002

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.9%

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

      1. unpow2 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.7 \lor \neg \left(im \leq 3.7\right):\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 7: 74.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 16500000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+151}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.3e+14)
   (* (* 0.5 re) (* im im))
   (if (<= im 16500000000000.0)
     (sin re)
     (if (<= im 2.9e+151)
       (+ re (* (pow re 3.0) -0.16666666666666666))
       (* (* 0.5 (sin re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.3e+14) {
		tmp = (0.5 * re) * (im * im);
	} else if (im <= 16500000000000.0) {
		tmp = sin(re);
	} else if (im <= 2.9e+151) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = (0.5 * sin(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 <= (-3.3d+14)) then
        tmp = (0.5d0 * re) * (im * im)
    else if (im <= 16500000000000.0d0) then
        tmp = sin(re)
    else if (im <= 2.9d+151) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.3e+14) {
		tmp = (0.5 * re) * (im * im);
	} else if (im <= 16500000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.9e+151) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.3e+14:
		tmp = (0.5 * re) * (im * im)
	elif im <= 16500000000000.0:
		tmp = math.sin(re)
	elif im <= 2.9e+151:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.3e+14)
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	elseif (im <= 16500000000000.0)
		tmp = sin(re);
	elseif (im <= 2.9e+151)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.3e+14)
		tmp = (0.5 * re) * (im * im);
	elseif (im <= 16500000000000.0)
		tmp = sin(re);
	elseif (im <= 2.9e+151)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.3e+14], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 16500000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.9e+151], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.3e14

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 49.8%

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

      1. unpow2 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in im around inf 49.8%

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

      1. associate-*r* 49.8%

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

      2. *-commutative 49.8%

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

      3. unpow2 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in re around 0 52.4%

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

       1. associate-*r* 52.4%

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

       2. unpow2 52.4%

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

   12. Simplified 52.4%

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

   if -3.3e14 < im < 1.65e13

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 95.5%

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

   if 1.65e13 < im < 2.90000000000000018e151

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.8%

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

      1. unpow2 4.8%

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

   6. Simplified 4.8%

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

   7. Taylor expanded in re around 0 7.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(2 + {im}^{2}\right) \cdot re\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 7.3%

         \[\leadsto \color{blue}{\left(\left(2 + {im}^{2}\right) \cdot re\right) \cdot 0.5} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      2. associate-*l* 7.3%

         \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(re \cdot 0.5\right)} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      3. metadata-eval 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(re \cdot \color{blue}{\left(-0.5 \cdot -1\right)}\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      4. associate-*l* 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \color{blue}{\left(\left(re \cdot -0.5\right) \cdot -1\right)} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      5. metadata-eval 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(\left(re \cdot \color{blue}{\left(-1 + 0.5\right)}\right) \cdot -1\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      6. distribute-rgt-out 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(\color{blue}{\left(-1 \cdot re + 0.5 \cdot re\right)} \cdot -1\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      7. *-commutative 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right)\right)} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      8. *-commutative 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right)\right) + \color{blue}{\left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \cdot -0.08333333333333333} \]

      9. associate-*l* 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right)\right) + \color{blue}{\left(2 + {im}^{2}\right) \cdot \left({re}^{3} \cdot -0.08333333333333333\right)} \]

      10. distribute-lft-out 29.0%

          \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right)} \]

      11. +-commutative 29.0%

          \[\leadsto \color{blue}{\left({im}^{2} + 2\right)} \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      12. unpow2 29.0%

          \[\leadsto \left(\color{blue}{im \cdot im} + 2\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      13. fma-def 29.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      14. distribute-rgt-out 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(-1 \cdot \color{blue}{\left(re \cdot \left(-1 + 0.5\right)\right)} + {re}^{3} \cdot -0.08333333333333333\right) \]

      15. metadata-eval 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(-1 \cdot \left(re \cdot \color{blue}{-0.5}\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      16. *-commutative 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(-1 \cdot \color{blue}{\left(-0.5 \cdot re\right)} + {re}^{3} \cdot -0.08333333333333333\right) \]

      17. associate-*r* 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(\color{blue}{\left(-1 \cdot -0.5\right) \cdot re} + {re}^{3} \cdot -0.08333333333333333\right) \]

      18. metadata-eval 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(\color{blue}{0.5} \cdot re + {re}^{3} \cdot -0.08333333333333333\right) \]

   9. Simplified 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re + {re}^{3} \cdot -0.08333333333333333\right)} \]

   10. Taylor expanded in im around 0 23.8%

       \[\leadsto \color{blue}{2 \cdot \left(0.5 \cdot re + -0.08333333333333333 \cdot {re}^{3}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 23.8%

          \[\leadsto \color{blue}{2 \cdot \left(0.5 \cdot re\right) + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right)} \]

       2. associate-*r* 23.8%

          \[\leadsto \color{blue}{\left(2 \cdot 0.5\right) \cdot re} + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right) \]

       3. metadata-eval 23.8%

          \[\leadsto \color{blue}{1} \cdot re + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right) \]

       4. *-lft-identity 23.8%

          \[\leadsto \color{blue}{re} + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right) \]

       5. *-commutative 23.8%

          \[\leadsto re + 2 \cdot \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right)} \]

       6. *-commutative 23.8%

          \[\leadsto re + \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right) \cdot 2} \]

       7. associate-*l* 23.8%

          \[\leadsto re + \color{blue}{{re}^{3} \cdot \left(-0.08333333333333333 \cdot 2\right)} \]

       8. metadata-eval 23.8%

          \[\leadsto re + {re}^{3} \cdot \color{blue}{-0.16666666666666666} \]

   12. Simplified 23.8%

       \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]

   if 2.90000000000000018e151 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.2%

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

      1. unpow2 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in im around inf 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. unpow2 97.2%

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

   9. Simplified 97.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 16500000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+151}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 71.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3200000000:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 16500000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.3 \cdot 10^{+150}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3200000000.0)
   (* (* 0.5 re) (* im im))
   (if (<= im 16500000000000.0)
     (sin re)
     (if (<= im 6.3e+150)
       (+ re (* (pow re 3.0) -0.16666666666666666))
       (* 0.5 (* re (+ 2.0 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3200000000.0) {
		tmp = (0.5 * re) * (im * im);
	} else if (im <= 16500000000000.0) {
		tmp = sin(re);
	} else if (im <= 6.3e+150) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = 0.5 * (re * (2.0 + (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 <= (-3200000000.0d0)) then
        tmp = (0.5d0 * re) * (im * im)
    else if (im <= 16500000000000.0d0) then
        tmp = sin(re)
    else if (im <= 6.3d+150) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3200000000.0) {
		tmp = (0.5 * re) * (im * im);
	} else if (im <= 16500000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.3e+150) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3200000000.0:
		tmp = (0.5 * re) * (im * im)
	elif im <= 16500000000000.0:
		tmp = math.sin(re)
	elif im <= 6.3e+150:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3200000000.0)
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	elseif (im <= 16500000000000.0)
		tmp = sin(re);
	elseif (im <= 6.3e+150)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3200000000.0)
		tmp = (0.5 * re) * (im * im);
	elseif (im <= 16500000000000.0)
		tmp = sin(re);
	elseif (im <= 6.3e+150)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3200000000.0], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 16500000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.3e+150], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.2e9

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 49.8%

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

      1. unpow2 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in im around inf 49.8%

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

      1. associate-*r* 49.8%

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

      2. *-commutative 49.8%

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

      3. unpow2 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in re around 0 52.4%

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

       1. associate-*r* 52.4%

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

       2. unpow2 52.4%

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

   12. Simplified 52.4%

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

   if -3.2e9 < im < 1.65e13

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 95.5%

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

   if 1.65e13 < im < 6.3000000000000003e150

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.8%

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

      1. unpow2 4.8%

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

   6. Simplified 4.8%

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

   7. Taylor expanded in re around 0 7.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(2 + {im}^{2}\right) \cdot re\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 7.3%

         \[\leadsto \color{blue}{\left(\left(2 + {im}^{2}\right) \cdot re\right) \cdot 0.5} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      2. associate-*l* 7.3%

         \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(re \cdot 0.5\right)} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      3. metadata-eval 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(re \cdot \color{blue}{\left(-0.5 \cdot -1\right)}\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      4. associate-*l* 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \color{blue}{\left(\left(re \cdot -0.5\right) \cdot -1\right)} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      5. metadata-eval 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(\left(re \cdot \color{blue}{\left(-1 + 0.5\right)}\right) \cdot -1\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      6. distribute-rgt-out 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(\color{blue}{\left(-1 \cdot re + 0.5 \cdot re\right)} \cdot -1\right) + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      7. *-commutative 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right)\right)} + -0.08333333333333333 \cdot \left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \]

      8. *-commutative 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right)\right) + \color{blue}{\left(\left(2 + {im}^{2}\right) \cdot {re}^{3}\right) \cdot -0.08333333333333333} \]

      9. associate-*l* 7.3%

         \[\leadsto \left(2 + {im}^{2}\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right)\right) + \color{blue}{\left(2 + {im}^{2}\right) \cdot \left({re}^{3} \cdot -0.08333333333333333\right)} \]

      10. distribute-lft-out 29.0%

          \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right)} \]

      11. +-commutative 29.0%

          \[\leadsto \color{blue}{\left({im}^{2} + 2\right)} \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      12. unpow2 29.0%

          \[\leadsto \left(\color{blue}{im \cdot im} + 2\right) \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      13. fma-def 29.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(-1 \cdot \left(-1 \cdot re + 0.5 \cdot re\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      14. distribute-rgt-out 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(-1 \cdot \color{blue}{\left(re \cdot \left(-1 + 0.5\right)\right)} + {re}^{3} \cdot -0.08333333333333333\right) \]

      15. metadata-eval 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(-1 \cdot \left(re \cdot \color{blue}{-0.5}\right) + {re}^{3} \cdot -0.08333333333333333\right) \]

      16. *-commutative 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(-1 \cdot \color{blue}{\left(-0.5 \cdot re\right)} + {re}^{3} \cdot -0.08333333333333333\right) \]

      17. associate-*r* 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(\color{blue}{\left(-1 \cdot -0.5\right) \cdot re} + {re}^{3} \cdot -0.08333333333333333\right) \]

      18. metadata-eval 29.0%

          \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(\color{blue}{0.5} \cdot re + {re}^{3} \cdot -0.08333333333333333\right) \]

   9. Simplified 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re + {re}^{3} \cdot -0.08333333333333333\right)} \]

   10. Taylor expanded in im around 0 23.8%

       \[\leadsto \color{blue}{2 \cdot \left(0.5 \cdot re + -0.08333333333333333 \cdot {re}^{3}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 23.8%

          \[\leadsto \color{blue}{2 \cdot \left(0.5 \cdot re\right) + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right)} \]

       2. associate-*r* 23.8%

          \[\leadsto \color{blue}{\left(2 \cdot 0.5\right) \cdot re} + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right) \]

       3. metadata-eval 23.8%

          \[\leadsto \color{blue}{1} \cdot re + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right) \]

       4. *-lft-identity 23.8%

          \[\leadsto \color{blue}{re} + 2 \cdot \left(-0.08333333333333333 \cdot {re}^{3}\right) \]

       5. *-commutative 23.8%

          \[\leadsto re + 2 \cdot \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right)} \]

       6. *-commutative 23.8%

          \[\leadsto re + \color{blue}{\left({re}^{3} \cdot -0.08333333333333333\right) \cdot 2} \]

       7. associate-*l* 23.8%

          \[\leadsto re + \color{blue}{{re}^{3} \cdot \left(-0.08333333333333333 \cdot 2\right)} \]

       8. metadata-eval 23.8%

          \[\leadsto re + {re}^{3} \cdot \color{blue}{-0.16666666666666666} \]

   12. Simplified 23.8%

       \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]

   if 6.3000000000000003e150 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.2%

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

      1. unpow2 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in re around 0 73.0%

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

      1. *-commutative 73.0%

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

      2. unpow2 73.0%

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

   9. Simplified 73.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3200000000:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 16500000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.3 \cdot 10^{+150}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 75.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (sin re)) (+ 2.0 (* im im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (2.0 + (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (2.0 + (im * im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (2.0 + (im * im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (2.0 + (im * im));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 77.9%

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

   1. unpow2 77.9%

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

6. Simplified 77.9%

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

7. Final simplification 77.9%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right) \]

Alternative 10: 70.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2050000000000:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+88}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2050000000000.0)
   (* (* 0.5 re) (* im im))
   (if (<= im 1.02e+88) (sin re) (* 0.5 (* re (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2050000000000.0) {
		tmp = (0.5 * re) * (im * im);
	} else if (im <= 1.02e+88) {
		tmp = sin(re);
	} else {
		tmp = 0.5 * (re * (2.0 + (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 <= (-2050000000000.0d0)) then
        tmp = (0.5d0 * re) * (im * im)
    else if (im <= 1.02d+88) then
        tmp = sin(re)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2050000000000.0) {
		tmp = (0.5 * re) * (im * im);
	} else if (im <= 1.02e+88) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2050000000000.0:
		tmp = (0.5 * re) * (im * im)
	elif im <= 1.02e+88:
		tmp = math.sin(re)
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2050000000000.0)
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	elseif (im <= 1.02e+88)
		tmp = sin(re);
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2050000000000.0)
		tmp = (0.5 * re) * (im * im);
	elseif (im <= 1.02e+88)
		tmp = sin(re);
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2050000000000.0], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+88], N[Sin[re], $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.05e12

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 49.8%

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

      1. unpow2 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in im around inf 49.8%

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

      1. associate-*r* 49.8%

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

      2. *-commutative 49.8%

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

      3. unpow2 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in re around 0 52.4%

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

       1. associate-*r* 52.4%

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

       2. unpow2 52.4%

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

   12. Simplified 52.4%

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

   if -2.05e12 < im < 1.01999999999999998e88

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 89.5%

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

   if 1.01999999999999998e88 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.4%

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

      1. unpow2 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in re around 0 53.8%

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

      1. *-commutative 53.8%

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

      2. unpow2 53.8%

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

   9. Simplified 53.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2050000000000:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+88}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 42.1% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -31000000 \lor \neg \left(im \leq 6.2\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -31000000.0) (not (<= im 6.2))) (* 0.5 (* im (* re im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -31000000.0) || !(im <= 6.2)) {
		tmp = 0.5 * (im * (re * im));
	} else {
		tmp = 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 <= (-31000000.0d0)) .or. (.not. (im <= 6.2d0))) then
        tmp = 0.5d0 * (im * (re * im))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -31000000.0) || !(im <= 6.2)) {
		tmp = 0.5 * (im * (re * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -31000000.0) or not (im <= 6.2):
		tmp = 0.5 * (im * (re * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -31000000.0) || !(im <= 6.2))
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -31000000.0) || ~((im <= 6.2)))
		tmp = 0.5 * (im * (re * im));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -31000000.0], N[Not[LessEqual[im, 6.2]], $MachinePrecision]], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -3.1e7 or 6.20000000000000018 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 54.0%

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

      1. unpow2 54.0%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in im around inf 54.0%

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

      1. associate-*r* 54.0%

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

      2. *-commutative 54.0%

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

      3. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in re around 0 48.1%

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

       1. *-commutative 48.1%

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

       2. unpow2 48.1%

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

       3. associate-*l* 30.1%

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

   12. Simplified 30.1%

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

   if -3.1e7 < im < 6.20000000000000018

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 52.5%

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

      1. associate-*r* 52.5%

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

      2. *-commutative 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in im around 0 51.6%

      \[\leadsto \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -31000000 \lor \neg \left(im \leq 6.2\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 12: 47.2% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -31000000 \lor \neg \left(im \leq 6.2\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -31000000.0) (not (<= im 6.2))) (* (* 0.5 re) (* im im)) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -31000000.0) || !(im <= 6.2)) {
		tmp = (0.5 * re) * (im * im);
	} else {
		tmp = 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 <= (-31000000.0d0)) .or. (.not. (im <= 6.2d0))) then
        tmp = (0.5d0 * re) * (im * im)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -31000000.0) || !(im <= 6.2)) {
		tmp = (0.5 * re) * (im * im);
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -31000000.0) or not (im <= 6.2):
		tmp = (0.5 * re) * (im * im)
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -31000000.0) || !(im <= 6.2))
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -31000000.0) || ~((im <= 6.2)))
		tmp = (0.5 * re) * (im * im);
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -31000000.0], N[Not[LessEqual[im, 6.2]], $MachinePrecision]], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -3.1e7 or 6.20000000000000018 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 54.0%

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

      1. unpow2 54.0%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in im around inf 54.0%

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

      1. associate-*r* 54.0%

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

      2. *-commutative 54.0%

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

      3. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in re around 0 48.1%

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

       1. associate-*r* 48.1%

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

       2. unpow2 48.1%

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

   12. Simplified 48.1%

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

   if -3.1e7 < im < 6.20000000000000018

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 52.5%

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

      1. associate-*r* 52.5%

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

      2. *-commutative 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in im around 0 51.6%

      \[\leadsto \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -31000000 \lor \neg \left(im \leq 6.2\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 13: 47.3% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* re (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	return 0.5 * (re * (2.0 + (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * (2.0d0 + (im * im)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * (2.0 + (im * im)));
}

Python

def code(re, im):
	return 0.5 * (re * (2.0 + (im * im)))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * (2.0 + (im * im)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 77.9%

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

   1. unpow2 77.9%

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

6. Simplified 77.9%

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

7. Taylor expanded in re around 0 50.3%

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

   1. *-commutative 50.3%

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

   2. unpow2 50.3%

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

9. Simplified 50.3%

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

10. Final simplification 50.3%

    \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right) \]

Alternative 14: 3.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1.9380669946781485 \cdot 10^{-10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.9380669946781485e-10)

C

double code(double re, double im) {
	return 1.9380669946781485e-10;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.9380669946781485d-10
end function

Java

public static double code(double re, double im) {
	return 1.9380669946781485e-10;
}

Python

def code(re, im):
	return 1.9380669946781485e-10

Julia

function code(re, im)
	return 1.9380669946781485e-10
end

MATLAB

function tmp = code(re, im)
	tmp = 1.9380669946781485e-10;
end

Wolfram

code[re_, im_] := 1.9380669946781485e-10

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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 91.5%

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

   1. associate-+r+ 91.5%

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

   2. unpow2 91.5%

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

6. Simplified 91.5%

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

8. Applied egg-rr 3.7%

   \[\leadsto \color{blue}{\frac{\sin re \cdot 1.9380669946781485 \cdot 10^{-10}}{\sin re + \left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10} - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}} \]
9. Step-by-step derivation

   1. *-commutative 3.7%

      \[\leadsto \frac{\color{blue}{1.9380669946781485 \cdot 10^{-10} \cdot \sin re}}{\sin re + \left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10} - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)} \]

   2. +-inverses 3.7%

      \[\leadsto \frac{1.9380669946781485 \cdot 10^{-10} \cdot \sin re}{\sin re + \color{blue}{0}} \]

   3. +-rgt-identity 3.7%

      \[\leadsto \frac{1.9380669946781485 \cdot 10^{-10} \cdot \sin re}{\color{blue}{\sin re}} \]

   4. associate-/l* 3.7%

      \[\leadsto \color{blue}{\frac{1.9380669946781485 \cdot 10^{-10}}{\frac{\sin re}{\sin re}}} \]

   5. *-inverses 3.7%

      \[\leadsto \frac{1.9380669946781485 \cdot 10^{-10}}{\color{blue}{1}} \]

   6. metadata-eval 3.7%

      \[\leadsto \color{blue}{1.9380669946781485 \cdot 10^{-10}} \]

10. Simplified 3.7%

    \[\leadsto \color{blue}{1.9380669946781485 \cdot 10^{-10}} \]

11. Final simplification 3.7%

    \[\leadsto 1.9380669946781485 \cdot 10^{-10} \]

Alternative 15: 4.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 5.0%

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

   1. +-inverses 5.0%

      \[\leadsto \frac{-2 \cdot \sin re}{-2 \cdot \sin re + \color{blue}{0}} \]

   2. +-rgt-identity 5.0%

      \[\leadsto \frac{-2 \cdot \sin re}{\color{blue}{-2 \cdot \sin re}} \]

   3. *-inverses 5.0%

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

7. Simplified 5.0%

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

8. Final simplification 5.0%

   \[\leadsto 1 \]

Alternative 16: 26.1% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 64.7%

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

   1. associate-*r* 64.7%

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

   2. *-commutative 64.7%

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

6. Simplified 64.7%

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

7. Taylor expanded in im around 0 29.8%

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

8. Final simplification 29.8%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023182 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))