math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 15

Speedup: 1.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return sin(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(sin(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 74.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 73.9%

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

5. Taylor expanded in re around inf 73.9%

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

   1. *-commutative 73.9%

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

7. Simplified 73.9%

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

8. Final simplification 73.9%

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

Alternative 4: 85.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left({re}^{3} \cdot -0.08333333333333333 + re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* 0.5 (exp im))))))
   (if (<= im 4.8)
     (* (* (sin re) 0.5) (+ 2.0 (* im im)))
     (if (<= im 3.1e+30)
       t_0
       (if (<= im 1.7e+40)
         (* (* im im) (+ (* (pow re 3.0) -0.08333333333333333) (* re 0.5)))
         (if (<= im 1.28e+103)
           t_0
           (*
            (sin re)
            (+ 1.0 (* im (+ 0.5 (* (* im im) 0.08333333333333333)))))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (0.5 * exp(im)));
	double tmp;
	if (im <= 4.8) {
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 3.1e+30) {
		tmp = t_0;
	} else if (im <= 1.7e+40) {
		tmp = (im * im) * ((pow(re, 3.0) * -0.08333333333333333) + (re * 0.5));
	} else if (im <= 1.28e+103) {
		tmp = t_0;
	} else {
		tmp = sin(re) * (1.0 + (im * (0.5 + ((im * im) * 0.08333333333333333))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (0.5d0 + (0.5d0 * exp(im)))
    if (im <= 4.8d0) then
        tmp = (sin(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 3.1d+30) then
        tmp = t_0
    else if (im <= 1.7d+40) then
        tmp = (im * im) * (((re ** 3.0d0) * (-0.08333333333333333d0)) + (re * 0.5d0))
    else if (im <= 1.28d+103) then
        tmp = t_0
    else
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 + ((im * im) * 0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (0.5 * Math.exp(im)));
	double tmp;
	if (im <= 4.8) {
		tmp = (Math.sin(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 3.1e+30) {
		tmp = t_0;
	} else if (im <= 1.7e+40) {
		tmp = (im * im) * ((Math.pow(re, 3.0) * -0.08333333333333333) + (re * 0.5));
	} else if (im <= 1.28e+103) {
		tmp = t_0;
	} else {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 + ((im * im) * 0.08333333333333333))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (0.5 * math.exp(im)))
	tmp = 0
	if im <= 4.8:
		tmp = (math.sin(re) * 0.5) * (2.0 + (im * im))
	elif im <= 3.1e+30:
		tmp = t_0
	elif im <= 1.7e+40:
		tmp = (im * im) * ((math.pow(re, 3.0) * -0.08333333333333333) + (re * 0.5))
	elif im <= 1.28e+103:
		tmp = t_0
	else:
		tmp = math.sin(re) * (1.0 + (im * (0.5 + ((im * im) * 0.08333333333333333))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))))
	tmp = 0.0
	if (im <= 4.8)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 3.1e+30)
		tmp = t_0;
	elseif (im <= 1.7e+40)
		tmp = Float64(Float64(im * im) * Float64(Float64((re ^ 3.0) * -0.08333333333333333) + Float64(re * 0.5)));
	elseif (im <= 1.28e+103)
		tmp = t_0;
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (0.5 * exp(im)));
	tmp = 0.0;
	if (im <= 4.8)
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 3.1e+30)
		tmp = t_0;
	elseif (im <= 1.7e+40)
		tmp = (im * im) * (((re ^ 3.0) * -0.08333333333333333) + (re * 0.5));
	elseif (im <= 1.28e+103)
		tmp = t_0;
	else
		tmp = sin(re) * (1.0 + (im * (0.5 + ((im * im) * 0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.8], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.1e+30], t$95$0, If[LessEqual[im, 1.7e+40], N[(N[(im * im), $MachinePrecision] * N[(N[(N[Power[re, 3.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision] + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.28e+103], t$95$0, N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 4.79999999999999982

   1. Initial program 100.0%

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

      1. 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 83.5%

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

      1. unpow2 83.5%

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

   6. Simplified 83.5%

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

   if 4.79999999999999982 < im < 3.0999999999999998e30 or 1.69999999999999994e40 < im < 1.28e103

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 80.0%

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

   if 3.0999999999999998e30 < im < 1.69999999999999994e40

   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 3.5%

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

      1. unpow2 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in im around inf 3.5%

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

      1. *-commutative 3.5%

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

      2. associate-*l* 3.5%

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

      3. *-commutative 3.5%

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

      4. unpow2 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in re around 0 40.6%

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

       1. unpow2 40.6%

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

       2. associate-*r* 40.6%

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

       3. unpow2 40.6%

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

       4. associate-*r* 40.6%

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

       5. distribute-rgt-out 80.6%

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

       6. *-commutative 80.6%

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

       7. *-commutative 80.6%

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

   12. Simplified 80.6%

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

   if 1.28e103 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(\left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right) + 0.25 \cdot {im}^{2}\right)}\right) \]

      2. associate-+l+ 100.0%

         \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)}\right) \]

      3. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(1 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)\right) \]

      4. unpow3 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*r* 100.0%

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

      7. unpow2 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. unpow2 100.0%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \left(0.08333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} + 0.25 \cdot im\right)\right)\right) \]

      12. associate-*r* 100.0%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \left(\color{blue}{\left(0.08333333333333333 \cdot im\right) \cdot im} + 0.25 \cdot im\right)\right)\right) \]

      13. distribute-rgt-out 100.0%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot \left(0.08333333333333333 \cdot im + 0.25\right)}\right)\right) \]

      14. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(\color{blue}{im \cdot 0.08333333333333333} + 0.25\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + 0.08333333333333333 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{0.08333333333333333 \cdot \left(im \cdot im\right)}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+30}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left({re}^{3} \cdot -0.08333333333333333 + re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+103}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 5: 84.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ t_1 := \left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left({re}^{3} \cdot -0.08333333333333333 + re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* 0.5 (exp im)))))
        (t_1 (* (* (sin re) 0.5) (+ 2.0 (* im im)))))
   (if (<= im 4.8)
     t_1
     (if (<= im 3.1e+30)
       t_0
       (if (<= im 1.7e+41)
         (* (* im im) (+ (* (pow re 3.0) -0.08333333333333333) (* re 0.5)))
         (if (<= im 1.3e+152) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (0.5 * exp(im)));
	double t_1 = (sin(re) * 0.5) * (2.0 + (im * im));
	double tmp;
	if (im <= 4.8) {
		tmp = t_1;
	} else if (im <= 3.1e+30) {
		tmp = t_0;
	} else if (im <= 1.7e+41) {
		tmp = (im * im) * ((pow(re, 3.0) * -0.08333333333333333) + (re * 0.5));
	} else if (im <= 1.3e+152) {
		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 = re * (0.5d0 + (0.5d0 * exp(im)))
    t_1 = (sin(re) * 0.5d0) * (2.0d0 + (im * im))
    if (im <= 4.8d0) then
        tmp = t_1
    else if (im <= 3.1d+30) then
        tmp = t_0
    else if (im <= 1.7d+41) then
        tmp = (im * im) * (((re ** 3.0d0) * (-0.08333333333333333d0)) + (re * 0.5d0))
    else if (im <= 1.3d+152) 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 = re * (0.5 + (0.5 * Math.exp(im)));
	double t_1 = (Math.sin(re) * 0.5) * (2.0 + (im * im));
	double tmp;
	if (im <= 4.8) {
		tmp = t_1;
	} else if (im <= 3.1e+30) {
		tmp = t_0;
	} else if (im <= 1.7e+41) {
		tmp = (im * im) * ((Math.pow(re, 3.0) * -0.08333333333333333) + (re * 0.5));
	} else if (im <= 1.3e+152) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (0.5 * math.exp(im)))
	t_1 = (math.sin(re) * 0.5) * (2.0 + (im * im))
	tmp = 0
	if im <= 4.8:
		tmp = t_1
	elif im <= 3.1e+30:
		tmp = t_0
	elif im <= 1.7e+41:
		tmp = (im * im) * ((math.pow(re, 3.0) * -0.08333333333333333) + (re * 0.5))
	elif im <= 1.3e+152:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))))
	t_1 = Float64(Float64(sin(re) * 0.5) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 4.8)
		tmp = t_1;
	elseif (im <= 3.1e+30)
		tmp = t_0;
	elseif (im <= 1.7e+41)
		tmp = Float64(Float64(im * im) * Float64(Float64((re ^ 3.0) * -0.08333333333333333) + Float64(re * 0.5)));
	elseif (im <= 1.3e+152)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (0.5 * exp(im)));
	t_1 = (sin(re) * 0.5) * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= 4.8)
		tmp = t_1;
	elseif (im <= 3.1e+30)
		tmp = t_0;
	elseif (im <= 1.7e+41)
		tmp = (im * im) * (((re ^ 3.0) * -0.08333333333333333) + (re * 0.5));
	elseif (im <= 1.3e+152)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.8], t$95$1, If[LessEqual[im, 3.1e+30], t$95$0, If[LessEqual[im, 1.7e+41], N[(N[(im * im), $MachinePrecision] * N[(N[(N[Power[re, 3.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision] + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3e+152], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.79999999999999982 or 1.3e152 < 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 85.1%

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

      1. unpow2 85.1%

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

   6. Simplified 85.1%

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

   if 4.79999999999999982 < im < 3.0999999999999998e30 or 1.69999999999999999e41 < im < 1.3e152

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 79.2%

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

   if 3.0999999999999998e30 < im < 1.69999999999999999e41

   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 3.5%

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

      1. unpow2 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in im around inf 3.5%

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

      1. *-commutative 3.5%

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

      2. associate-*l* 3.5%

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

      3. *-commutative 3.5%

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

      4. unpow2 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in re around 0 40.6%

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

       1. unpow2 40.6%

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

       2. associate-*r* 40.6%

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

       3. unpow2 40.6%

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

       4. associate-*r* 40.6%

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

       5. distribute-rgt-out 80.6%

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

       6. *-commutative 80.6%

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

       7. *-commutative 80.6%

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

   12. Simplified 80.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+30}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left({re}^{3} \cdot -0.08333333333333333 + re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 6: 84.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.5 \lor \neg \left(im \leq 1.3 \cdot 10^{+152}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 3.5) (not (<= im 1.3e+152)))
   (* (* (sin re) 0.5) (+ 2.0 (* im im)))
   (* re (+ 0.5 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 3.5) || !(im <= 1.3e+152)) {
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = re * (0.5 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 3.5d0) .or. (.not. (im <= 1.3d+152))) then
        tmp = (sin(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 3.5) || !(im <= 1.3e+152)) {
		tmp = (Math.sin(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 3.5) or not (im <= 1.3e+152):
		tmp = (math.sin(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 3.5) || !(im <= 1.3e+152))
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 3.5) || ~((im <= 1.3e+152)))
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = re * (0.5 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 3.5], N[Not[LessEqual[im, 1.3e+152]], $MachinePrecision]], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.5 or 1.3e152 < 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 85.1%

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

      1. unpow2 85.1%

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

   6. Simplified 85.1%

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

   if 3.5 < im < 1.3e152

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 69.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5 \lor \neg \left(im \leq 1.3 \cdot 10^{+152}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 7: 68.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7e+40)
   (sin re)
   (if (<= im 1.3e+152)
     (* 0.041666666666666664 (* re (pow im 4.0)))
     (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.7e+40) {
		tmp = sin(re);
	} else if (im <= 1.3e+152) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else {
		tmp = sin(re) * (0.5 * (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 <= 1.7d+40) then
        tmp = sin(re)
    else if (im <= 1.3d+152) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.7e+40) {
		tmp = Math.sin(re);
	} else if (im <= 1.3e+152) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.7e+40:
		tmp = math.sin(re)
	elif im <= 1.3e+152:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.7e+40)
		tmp = sin(re);
	elseif (im <= 1.3e+152)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.7e+40)
		tmp = sin(re);
	elseif (im <= 1.3e+152)
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.7e+40], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.3e+152], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.69999999999999994e40

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 64.0%

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

   if 1.69999999999999994e40 < im < 1.3e152

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 80.0%

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

      1. distribute-lft-out 80.0%

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

      2. +-commutative 80.0%

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

      3. rec-exp 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in im around 0 70.7%

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

      1. unpow2 70.7%

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

   9. Simplified 70.7%

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

   10. Taylor expanded in im around inf 70.7%

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

   if 1.3e152 < 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 94.6%

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

      1. unpow2 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in im around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l* 94.6%

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

      3. *-commutative 94.6%

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

      4. unpow2 94.6%

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

   9. Simplified 94.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 72.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.5:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.5)
   (sin re)
   (if (<= im 1.3e+152)
     (* re (+ 0.5 (* 0.5 (exp im))))
     (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.5) {
		tmp = sin(re);
	} else if (im <= 1.3e+152) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = sin(re) * (0.5 * (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 <= 5.5d0) then
        tmp = sin(re)
    else if (im <= 1.3d+152) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.5) {
		tmp = Math.sin(re);
	} else if (im <= 1.3e+152) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.5:
		tmp = math.sin(re)
	elif im <= 1.3e+152:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.5)
		tmp = sin(re);
	elseif (im <= 1.3e+152)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.5)
		tmp = sin(re);
	elseif (im <= 1.3e+152)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.5], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.3e+152], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.5

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.8%

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

   if 5.5 < im < 1.3e152

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 69.0%

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

   if 1.3e152 < 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 94.6%

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

      1. unpow2 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in im around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l* 94.6%

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

      3. *-commutative 94.6%

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

      4. unpow2 94.6%

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

   9. Simplified 94.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.5:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 65.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7e+40) (sin re) (* 0.041666666666666664 (* re (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.7e+40) {
		tmp = sin(re);
	} else {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.7d+40) then
        tmp = sin(re)
    else
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.7e+40) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.7e+40:
		tmp = math.sin(re)
	else:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.7e+40)
		tmp = sin(re);
	else
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.7e+40)
		tmp = sin(re);
	else
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.7e+40], N[Sin[re], $MachinePrecision], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.69999999999999994e40

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 64.0%

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

   if 1.69999999999999994e40 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.9%

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

      1. distribute-lft-out 76.9%

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

      2. +-commutative 76.9%

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

      3. rec-exp 76.9%

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

   6. Simplified 76.9%

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

   7. Taylor expanded in im around 0 73.3%

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

      1. unpow2 73.3%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in im around inf 73.3%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 10: 64.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7e+40)
   (sin re)
   (* re (+ 1.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.7e+40) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.7d+40) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.7e+40) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.7e+40:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.7e+40)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.7e+40)
		tmp = sin(re);
	else
		tmp = re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.7e+40], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.69999999999999994e40

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 64.0%

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

   if 1.69999999999999994e40 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in im around 0 79.9%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 79.9%

         \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(\left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right) + 0.25 \cdot {im}^{2}\right)}\right) \]

      2. associate-+l+ 79.9%

         \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)}\right) \]

      3. *-commutative 79.9%

         \[\leadsto \sin re \cdot \left(1 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)\right) \]

      4. unpow3 79.9%

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

      5. unpow2 79.9%

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

      6. associate-*r* 79.9%

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

      7. unpow2 79.9%

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

      8. associate-*r* 79.9%

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

      9. distribute-rgt-out 79.9%

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

      10. distribute-lft-out 79.9%

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

      11. unpow2 79.9%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \left(0.08333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} + 0.25 \cdot im\right)\right)\right) \]

      12. associate-*r* 79.9%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \left(\color{blue}{\left(0.08333333333333333 \cdot im\right) \cdot im} + 0.25 \cdot im\right)\right)\right) \]

      13. distribute-rgt-out 79.9%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot \left(0.08333333333333333 \cdot im + 0.25\right)}\right)\right) \]

      14. *-commutative 79.9%

          \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(\color{blue}{im \cdot 0.08333333333333333} + 0.25\right)\right)\right) \]

   7. Simplified 79.9%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]

   8. Taylor expanded in re around 0 69.6%

      \[\leadsto \color{blue}{\left(\left(\left(0.25 + 0.08333333333333333 \cdot im\right) \cdot im + 0.5\right) \cdot im + 1\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \]

Alternative 11: 44.7% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* re (+ 1.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	return re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
}

Python

def code(re, im):
	return re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 73.9%

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

5. Taylor expanded in im around 0 65.8%

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 65.8%

      \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(\left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right) + 0.25 \cdot {im}^{2}\right)}\right) \]

   2. associate-+l+ 65.8%

      \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)}\right) \]

   3. *-commutative 65.8%

      \[\leadsto \sin re \cdot \left(1 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)\right) \]

   4. unpow3 65.8%

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

   5. unpow2 65.8%

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

   6. associate-*r* 65.8%

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

   7. unpow2 65.8%

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

   8. associate-*r* 65.8%

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

   9. distribute-rgt-out 65.8%

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

   10. distribute-lft-out 65.8%

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

   11. unpow2 65.8%

       \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \left(0.08333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} + 0.25 \cdot im\right)\right)\right) \]

   12. associate-*r* 65.8%

       \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \left(\color{blue}{\left(0.08333333333333333 \cdot im\right) \cdot im} + 0.25 \cdot im\right)\right)\right) \]

   13. distribute-rgt-out 65.8%

       \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot \left(0.08333333333333333 \cdot im + 0.25\right)}\right)\right) \]

   14. *-commutative 65.8%

       \[\leadsto \sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(\color{blue}{im \cdot 0.08333333333333333} + 0.25\right)\right)\right) \]

7. Simplified 65.8%

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]

8. Taylor expanded in re around 0 42.6%

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

9. Final simplification 42.6%

   \[\leadsto re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right) \]

Alternative 12: 37.1% accurate, 34.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = re
    else
        tmp = (im * im) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 50.5%

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

      1. distribute-lft-out 50.5%

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

      2. +-commutative 50.5%

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

      3. rec-exp 50.5%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in im around 0 28.7%

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

   if 1.3999999999999999 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 72.1%

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

   5. Taylor expanded in im around 0 47.2%

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

      1. unpow2 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in im around inf 47.2%

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

      1. unpow2 47.2%

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

      2. associate-*r* 47.2%

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

      3. *-commutative 47.2%

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

      4. *-commutative 47.2%

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

   10. Simplified 47.2%

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

4. Final simplification 33.1%

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

Alternative 13: 47.8% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re * (1.0 + (0.5 * (im * im)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 55.6%

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

5. Taylor expanded in im around 0 43.1%

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

   1. unpow2 43.1%

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

7. Simplified 43.1%

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

8. Final simplification 43.1%

   \[\leadsto re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

Alternative 14: 47.8% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 55.6%

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

5. Taylor expanded in im around 0 43.1%

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

   1. unpow2 43.1%

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

7. Simplified 43.1%

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

   1. +-commutative 43.1%

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

   2. distribute-rgt-in 43.1%

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

   3. *-commutative 43.1%

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

   4. associate-*l* 43.1%

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

   5. *-un-lft-identity 43.1%

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

9. Applied egg-rr 43.1%

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

10. Final simplification 43.1%

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

Alternative 15: 26.4% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 55.6%

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

   1. distribute-lft-out 55.6%

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

   2. +-commutative 55.6%

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

   3. rec-exp 55.6%

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

6. Simplified 55.6%

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

7. Taylor expanded in im around 0 22.5%

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

8. Final simplification 22.5%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023230 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))