math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 15

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot 0.75\\ \mathbf{if}\;im \leq 78:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) 0.75)))
   (if (<= im 78.0)
     (* (sin re) (fma im (* 0.5 im) 1.0))
     (if (<= im 1.5e+114)
       t_0
       (if (<= im 2.3e+144)
         (log1p (expm1 re))
         (if (<= im 1.35e+154) t_0 (* (* 0.5 (sin re)) (* im im))))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * 0.75;
	double tmp;
	if (im <= 78.0) {
		tmp = sin(re) * fma(im, (0.5 * im), 1.0);
	} else if (im <= 1.5e+114) {
		tmp = t_0;
	} else if (im <= 2.3e+144) {
		tmp = log1p(expm1(re));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * 0.75)
	tmp = 0.0
	if (im <= 78.0)
		tmp = Float64(sin(re) * fma(im, Float64(0.5 * im), 1.0));
	elseif (im <= 1.5e+114)
		tmp = t_0;
	elseif (im <= 2.3e+144)
		tmp = log1p(expm1(re));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * 0.75), $MachinePrecision]}, If[LessEqual[im, 78.0], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.5e+114], t$95$0, If[LessEqual[im, 2.3e+144], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 78

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 59.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - 1\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 59.0%

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

   8. Taylor expanded in im around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*r* 82.9%

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

      3. distribute-rgt1-in 82.9%

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

      4. unpow2 82.9%

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

      5. associate-*r* 82.9%

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

      6. *-commutative 82.9%

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

      7. fma-udef 82.9%

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

   10. Simplified 82.9%

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

   if 78 < im < 1.5e114 or 2.3000000000000001e144 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 50.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - 1\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 50.0%

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

   9. Applied egg-rr 38.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{1.5}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   if 1.5e114 < im < 2.3000000000000001e144

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 100.0%

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

   6. Applied egg-rr 58.1%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 78:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+114}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 3: 84.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e-5)
   (* (sin re) (fma im (* 0.5 im) 1.0))
   (if (<= im 2.3e+151)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (* 0.5 (sin re)) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e-5) {
		tmp = sin(re) * fma(im, (0.5 * im), 1.0);
	} else if (im <= 2.3e+151) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e-5)
		tmp = Float64(sin(re) * fma(im, Float64(0.5 * im), 1.0));
	elseif (im <= 2.3e+151)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e-5], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e+151], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.00000000000000057e-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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 59.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - 1\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 59.3%

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

   8. Taylor expanded in im around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*r* 82.8%

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

      3. distribute-rgt1-in 82.8%

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

      4. unpow2 82.8%

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

      5. associate-*r* 82.8%

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

      6. *-commutative 82.8%

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

      7. fma-udef 82.8%

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

   10. Simplified 82.8%

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

   if 9.00000000000000057e-5 < im < 2.3000000000000001e151

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.8%

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

   if 2.3000000000000001e151 < 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 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in re around 0 96.9%

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

   8. Taylor expanded in im around inf 96.9%

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

      1. associate-*r* 96.9%

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

      2. unpow2 96.9%

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

   10. Simplified 96.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 4: 77.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 155000000:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+113}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot \left(re + {re}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 155000000.0)
   (* (sin re) (fma im (* 0.5 im) 1.0))
   (if (<= im 1.02e+113)
     (+ re (* (* im im) (* 0.5 (+ re (* (pow re 3.0) -0.16666666666666666)))))
     (if (<= im 2.3e+151) (log1p (expm1 re)) (* (* 0.5 (sin re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 155000000.0) {
		tmp = sin(re) * fma(im, (0.5 * im), 1.0);
	} else if (im <= 1.02e+113) {
		tmp = re + ((im * im) * (0.5 * (re + (pow(re, 3.0) * -0.16666666666666666))));
	} else if (im <= 2.3e+151) {
		tmp = log1p(expm1(re));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 155000000.0)
		tmp = Float64(sin(re) * fma(im, Float64(0.5 * im), 1.0));
	elseif (im <= 1.02e+113)
		tmp = Float64(re + Float64(Float64(im * im) * Float64(0.5 * Float64(re + Float64((re ^ 3.0) * -0.16666666666666666)))));
	elseif (im <= 2.3e+151)
		tmp = log1p(expm1(re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 155000000.0], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.02e+113], N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e+151], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.55e8

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 58.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - 1\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 58.9%

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

   8. Taylor expanded in im around 0 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r* 82.1%

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

      3. distribute-rgt1-in 82.1%

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

      4. unpow2 82.1%

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

      5. associate-*r* 82.1%

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

      6. *-commutative 82.1%

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

      7. fma-udef 82.1%

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

   10. Simplified 82.1%

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

   if 1.55e8 < im < 1.02000000000000002e113

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in re around 0 2.8%

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

   8. Taylor expanded in re around 0 37.0%

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

   10. Simplified 37.0%

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

   if 1.02000000000000002e113 < im < 2.3000000000000001e151

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 100.0%

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

   6. Applied egg-rr 58.1%

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

   if 2.3000000000000001e151 < 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 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in re around 0 96.9%

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

   8. Taylor expanded in im around inf 96.9%

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

      1. associate-*r* 96.9%

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

      2. unpow2 96.9%

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

   10. Simplified 96.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 155000000:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+113}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot \left(re + {re}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 5: 64.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+113}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot \left(re + {re}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e-5)
   (sin re)
   (if (<= im 4.3e+113)
     (+ re (* (* im im) (* 0.5 (+ re (* (pow re 3.0) -0.16666666666666666)))))
     (if (<= im 2.3e+151) (log1p (expm1 re)) (* (* 0.5 (sin re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e-5) {
		tmp = sin(re);
	} else if (im <= 4.3e+113) {
		tmp = re + ((im * im) * (0.5 * (re + (pow(re, 3.0) * -0.16666666666666666))));
	} else if (im <= 2.3e+151) {
		tmp = log1p(expm1(re));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9e-5) {
		tmp = Math.sin(re);
	} else if (im <= 4.3e+113) {
		tmp = re + ((im * im) * (0.5 * (re + (Math.pow(re, 3.0) * -0.16666666666666666))));
	} else if (im <= 2.3e+151) {
		tmp = Math.log1p(Math.expm1(re));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9e-5:
		tmp = math.sin(re)
	elif im <= 4.3e+113:
		tmp = re + ((im * im) * (0.5 * (re + (math.pow(re, 3.0) * -0.16666666666666666))))
	elif im <= 2.3e+151:
		tmp = math.log1p(math.expm1(re))
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e-5)
		tmp = sin(re);
	elseif (im <= 4.3e+113)
		tmp = Float64(re + Float64(Float64(im * im) * Float64(0.5 * Float64(re + Float64((re ^ 3.0) * -0.16666666666666666)))));
	elseif (im <= 2.3e+151)
		tmp = log1p(expm1(re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e-5], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.3e+113], N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e+151], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.00000000000000057e-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. 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 68.4%

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

   if 9.00000000000000057e-5 < im < 4.3000000000000003e113

   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 7.7%

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

   6. Simplified 7.7%

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

   7. Taylor expanded in re around 0 6.6%

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

   8. Taylor expanded in re around 0 36.7%

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

   10. Simplified 36.7%

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

   if 4.3000000000000003e113 < im < 2.3000000000000001e151

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 100.0%

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

   6. Applied egg-rr 58.1%

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

   if 2.3000000000000001e151 < 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 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in re around 0 96.9%

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

   8. Taylor expanded in im around inf 96.9%

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

      1. associate-*r* 96.9%

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

      2. unpow2 96.9%

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

   10. Simplified 96.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+113}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot \left(re + {re}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 6: 64.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+112}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot \left(re + {re}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e-5)
   (sin re)
   (if (<= im 5e+112)
     (+ re (* (* im im) (* 0.5 (+ re (* (pow re 3.0) -0.16666666666666666)))))
     (if (<= im 2.3e+151)
       (* im (* re (* 0.5 im)))
       (* (* 0.5 (sin re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e-5) {
		tmp = sin(re);
	} else if (im <= 5e+112) {
		tmp = re + ((im * im) * (0.5 * (re + (pow(re, 3.0) * -0.16666666666666666))));
	} else if (im <= 2.3e+151) {
		tmp = im * (re * (0.5 * im));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9d-5) then
        tmp = sin(re)
    else if (im <= 5d+112) then
        tmp = re + ((im * im) * (0.5d0 * (re + ((re ** 3.0d0) * (-0.16666666666666666d0)))))
    else if (im <= 2.3d+151) then
        tmp = im * (re * (0.5d0 * im))
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9e-5) {
		tmp = Math.sin(re);
	} else if (im <= 5e+112) {
		tmp = re + ((im * im) * (0.5 * (re + (Math.pow(re, 3.0) * -0.16666666666666666))));
	} else if (im <= 2.3e+151) {
		tmp = im * (re * (0.5 * im));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9e-5:
		tmp = math.sin(re)
	elif im <= 5e+112:
		tmp = re + ((im * im) * (0.5 * (re + (math.pow(re, 3.0) * -0.16666666666666666))))
	elif im <= 2.3e+151:
		tmp = im * (re * (0.5 * im))
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e-5)
		tmp = sin(re);
	elseif (im <= 5e+112)
		tmp = Float64(re + Float64(Float64(im * im) * Float64(0.5 * Float64(re + Float64((re ^ 3.0) * -0.16666666666666666)))));
	elseif (im <= 2.3e+151)
		tmp = Float64(im * Float64(re * Float64(0.5 * im)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9e-5)
		tmp = sin(re);
	elseif (im <= 5e+112)
		tmp = re + ((im * im) * (0.5 * (re + ((re ^ 3.0) * -0.16666666666666666))));
	elseif (im <= 2.3e+151)
		tmp = im * (re * (0.5 * im));
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e-5], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5e+112], N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e+151], N[(im * N[(re * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.00000000000000057e-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. 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 68.4%

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

   if 9.00000000000000057e-5 < im < 5e112

   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 7.8%

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

   6. Simplified 7.8%

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

   7. Taylor expanded in re around 0 6.6%

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

   8. Taylor expanded in re around 0 38.0%

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

   10. Simplified 38.0%

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

   if 5e112 < im < 2.3000000000000001e151

   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 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 40.7%

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

   8. Taylor expanded in im around 0 40.7%

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

      1. unpow2 40.7%

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

   10. Simplified 40.7%

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

   11. Taylor expanded in im around inf 40.7%

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

       1. *-commutative 40.7%

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

       2. associate-*r* 40.7%

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

       3. *-commutative 40.7%

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

       4. unpow2 40.7%

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

       5. associate-*l* 40.7%

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

       6. associate-*l* 40.7%

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

       7. *-commutative 40.7%

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

   13. Simplified 40.7%

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

   if 2.3000000000000001e151 < 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 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in re around 0 96.9%

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

   8. Taylor expanded in im around inf 96.9%

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

      1. associate-*r* 96.9%

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

      2. unpow2 96.9%

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

   10. Simplified 96.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+112}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot \left(re + {re}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 64.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 220000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+96}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 220000000000.0)
   (sin re)
   (if (<= im 8.2e+96)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (if (<= im 2.3e+151)
       (* im (* re (* 0.5 im)))
       (* (* 0.5 (sin re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 220000000000.0) {
		tmp = sin(re);
	} else if (im <= 8.2e+96) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else if (im <= 2.3e+151) {
		tmp = im * (re * (0.5 * im));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 220000000000.0d0) then
        tmp = sin(re)
    else if (im <= 8.2d+96) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else if (im <= 2.3d+151) then
        tmp = im * (re * (0.5d0 * im))
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 220000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 8.2e+96) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else if (im <= 2.3e+151) {
		tmp = im * (re * (0.5 * im));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 220000000000.0:
		tmp = math.sin(re)
	elif im <= 8.2e+96:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	elif im <= 2.3e+151:
		tmp = im * (re * (0.5 * im))
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 220000000000.0)
		tmp = sin(re);
	elseif (im <= 8.2e+96)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	elseif (im <= 2.3e+151)
		tmp = Float64(im * Float64(re * Float64(0.5 * im)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 220000000000.0)
		tmp = sin(re);
	elseif (im <= 8.2e+96)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	elseif (im <= 2.3e+151)
		tmp = im * (re * (0.5 * im));
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 220000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 8.2e+96], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e+151], N[(im * N[(re * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.2e11

   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 67.7%

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

   if 2.2e11 < im < 8.19999999999999996e96

   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 2.7%

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

   5. Taylor expanded in re around 0 39.0%

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

   7. Simplified 39.0%

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

   if 8.19999999999999996e96 < im < 2.3000000000000001e151

   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 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in re around 0 31.4%

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

   8. Taylor expanded in im around 0 31.4%

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

      1. unpow2 31.4%

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

   10. Simplified 31.4%

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

   11. Taylor expanded in im around inf 31.4%

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

       1. *-commutative 31.4%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

       4. unpow2 31.4%

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

       5. associate-*l* 31.4%

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

       6. associate-*l* 31.4%

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

       7. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if 2.3000000000000001e151 < 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 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in re around 0 96.9%

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

   8. Taylor expanded in im around inf 96.9%

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

      1. associate-*r* 96.9%

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

      2. unpow2 96.9%

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

   10. Simplified 96.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 220000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+96}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 61.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 190000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 190000000000.0)
   (sin re)
   (if (<= im 5.8e+97)
     (+ re (* (pow re 3.0) -0.16666666666666666))
     (* (* 0.5 re) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 190000000000.0) {
		tmp = sin(re);
	} else if (im <= 5.8e+97) {
		tmp = re + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 190000000000.0d0) then
        tmp = sin(re)
    else if (im <= 5.8d+97) then
        tmp = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = (0.5d0 * re) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 190000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 5.8e+97) {
		tmp = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 190000000000.0:
		tmp = math.sin(re)
	elif im <= 5.8e+97:
		tmp = re + (math.pow(re, 3.0) * -0.16666666666666666)
	else:
		tmp = (0.5 * re) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 190000000000.0)
		tmp = sin(re);
	elseif (im <= 5.8e+97)
		tmp = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 190000000000.0)
		tmp = sin(re);
	elseif (im <= 5.8e+97)
		tmp = re + ((re ^ 3.0) * -0.16666666666666666);
	else
		tmp = (0.5 * re) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 190000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5.8e+97], N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.9e11

   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 67.7%

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

   if 1.9e11 < im < 5.79999999999999974e97

   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 2.7%

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

   5. Taylor expanded in re around 0 39.0%

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

   7. Simplified 39.0%

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

   if 5.79999999999999974e97 < 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 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in re around 0 58.1%

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

   8. Taylor expanded in im around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*l* 58.1%

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

      4. unpow2 58.1%

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

   10. Simplified 58.1%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 190000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 9: 61.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 155000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 155000000.0)
   (sin re)
   (if (<= im 1.6e+104) (pow re -512.0) (* (* 0.5 re) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 155000000.0) {
		tmp = sin(re);
	} else if (im <= 1.6e+104) {
		tmp = pow(re, -512.0);
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 155000000.0d0) then
        tmp = sin(re)
    else if (im <= 1.6d+104) then
        tmp = re ** (-512.0d0)
    else
        tmp = (0.5d0 * re) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 155000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.6e+104) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 155000000.0:
		tmp = math.sin(re)
	elif im <= 1.6e+104:
		tmp = math.pow(re, -512.0)
	else:
		tmp = (0.5 * re) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 155000000.0)
		tmp = sin(re);
	elseif (im <= 1.6e+104)
		tmp = re ^ -512.0;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 155000000.0)
		tmp = sin(re);
	elseif (im <= 1.6e+104)
		tmp = re ^ -512.0;
	else
		tmp = (0.5 * re) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 155000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.6e+104], N[Power[re, -512.0], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.55e8

   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 67.7%

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

   if 1.55e8 < im < 1.6e104

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 64.7%

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

   6. Applied egg-rr 24.1%

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

   if 1.6e104 < 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 67.9%

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

   6. Simplified 67.9%

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

   7. Taylor expanded in re around 0 59.4%

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

   8. Taylor expanded in im around inf 59.4%

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

      1. *-commutative 59.4%

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

      2. *-commutative 59.4%

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

      3. associate-*l* 59.4%

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

      4. unpow2 59.4%

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

   10. Simplified 59.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 155000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 60.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2e-5) (sin re) (* re (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2e-5) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + (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 <= 6.2d-5) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2e-5) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2e-5:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + (0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2e-5)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2e-5)
		tmp = sin(re);
	else
		tmp = re * (1.0 + (0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2e-5], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.20000000000000027e-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. 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 68.4%

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

   if 6.20000000000000027e-5 < im

   1. Initial program 100.0%

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

      1. 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 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in re around 0 42.3%

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

   8. Taylor expanded in im around 0 42.3%

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

      1. unpow2 42.3%

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

   10. Simplified 42.3%

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

   11. Taylor expanded in re around 0 42.3%

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

       1. *-commutative 42.3%

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

       2. unpow2 42.3%

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

   13. Simplified 42.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 47.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+121}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e+121) (* re re) (* re (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+121) {
		tmp = re * re;
	} else {
		tmp = re * (1.0 + (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 (re <= (-3.4d+121)) then
        tmp = re * re
    else
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+121) {
		tmp = re * re;
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e+121:
		tmp = re * re
	else:
		tmp = re * (1.0 + (0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e+121)
		tmp = Float64(re * re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e+121)
		tmp = re * re;
	else
		tmp = re * (1.0 + (0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e+121], N[(re * re), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.4000000000000001e121

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 13.0%

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

   6. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{\sqrt[3]{re}} \]

   8. Applied egg-rr 27.0%

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

   if -3.4000000000000001e121 < re

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in re around 0 49.5%

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

   8. Taylor expanded in im around 0 49.5%

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

      1. unpow2 49.5%

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

   10. Simplified 49.5%

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

   11. Taylor expanded in re around 0 49.5%

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

       1. *-commutative 49.5%

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

       2. unpow2 49.5%

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

   13. Simplified 49.5%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+121}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 12: 33.6% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.45) re (* im (* re (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = re;
	} else {
		tmp = im * (re * (0.5 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.45d0) then
        tmp = re
    else
        tmp = im * (re * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = re;
	} else {
		tmp = im * (re * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.45:
		tmp = re
	else:
		tmp = im * (re * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.45)
		tmp = re;
	else
		tmp = Float64(im * Float64(re * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.45)
		tmp = re;
	else
		tmp = im * (re * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.45], re, N[(im * N[(re * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.44999999999999996

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 53.8%

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

   5. Taylor expanded in im around 0 30.9%

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

   if 1.44999999999999996 < 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 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in re around 0 41.3%

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

   8. Taylor expanded in im around 0 41.3%

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

      1. unpow2 41.3%

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

   10. Simplified 41.3%

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

   11. Taylor expanded in im around inf 41.3%

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

       1. *-commutative 41.3%

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

       2. associate-*r* 41.3%

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

       3. *-commutative 41.3%

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

       4. unpow2 41.3%

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

       5. associate-*l* 41.3%

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

       6. associate-*l* 32.0%

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

       7. *-commutative 32.0%

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

   13. Simplified 32.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 13: 36.4% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.45) re (* (* 0.5 re) (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.45d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.45) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.45:
		tmp = re
	else:
		tmp = (0.5 * re) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.45)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.45)
		tmp = re;
	else
		tmp = (0.5 * re) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.45], re, N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.44999999999999996

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 53.8%

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

   5. Taylor expanded in im around 0 30.9%

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

   if 1.44999999999999996 < 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 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in re around 0 41.3%

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

   8. Taylor expanded in im around inf 41.3%

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

      1. *-commutative 41.3%

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

      2. *-commutative 41.3%

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

      3. associate-*l* 41.3%

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

      4. unpow2 41.3%

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

   10. Simplified 41.3%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.45:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 14: 29.0% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.35 \cdot 10^{+19}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 2.35e+19) re (* re re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.35e+19) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.35d+19) then
        tmp = re
    else
        tmp = re * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.35e+19) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.35e+19:
		tmp = re
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.35e+19)
		tmp = re;
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.35e+19)
		tmp = re;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.35e+19], re, N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.35e19

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 54.4%

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

   5. Taylor expanded in im around 0 30.2%

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

   if 2.35e19 < 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 re around 0 72.7%

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

   6. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\sqrt[3]{re}} \]

   8. Applied egg-rr 12.5%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.35 \cdot 10^{+19}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 15: 26.3% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 58.3%

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

5. Taylor expanded in im around 0 24.2%

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

6. Final simplification 24.2%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023201 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))