math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 16

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 84.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.115 \lor \neg \left(im \leq 1.15 \cdot 10^{+146}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.115) (not (<= im 1.15e+146)))
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (* (+ (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.115) || !(im <= 1.15e+146)) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.115) || !(im <= 1.15e+146))
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.115], N[Not[LessEqual[im, 1.15e+146]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.115000000000000005 or 1.15e146 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 88.4%

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

   7. Simplified 88.4%

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

   if 0.115000000000000005 < im < 1.15e146

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.115 \lor \neg \left(im \leq 1.15 \cdot 10^{+146}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 700.0) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (log (/ -2.0 (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 700.0) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else {
		tmp = log((-2.0 / exp(re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 700.0) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	else
		tmp = log(Float64(-2.0 / exp(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 700.0], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(-2.0 / N[Exp[re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 700 or 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 88.8%

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

   7. Simplified 88.8%

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

   if 700 < 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 13.6%

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

   7. Taylor expanded in re around 0 13.5%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]

   9. Applied egg-rr 15.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0)
   (sin re)
   (if (<= im 2e+119) (log (/ -2.0 (exp re))) (* re (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = sin(re);
	} else if (im <= 2e+119) {
		tmp = log((-2.0 / exp(re)));
	} else {
		tmp = re * (0.5 * fma(im, im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = sin(re);
	elseif (im <= 2e+119)
		tmp = log(Float64(-2.0 / exp(re)));
	else
		tmp = Float64(re * Float64(0.5 * fma(im, im, 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2e+119], N[Log[N[(-2.0 / N[Exp[re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(re * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 720

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 70.2%

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

   if 720 < im < 1.99999999999999989e119

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 12.9%

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

   7. Taylor expanded in re around 0 12.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]

   9. Applied egg-rr 19.2%

      \[\leadsto \color{blue}{\log \left(\frac{-2}{e^{re}}\right)} \]

   if 1.99999999999999989e119 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in re around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*r* 64.3%

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

      3. +-commutative 64.3%

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

      4. unpow2 64.3%

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

      5. fma-udef 64.3%

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

   10. Simplified 64.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12400000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12400000.0)
   (sin re)
   (if (<= im 2e+119) (pow re -2.0) (* re (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12400000.0) {
		tmp = sin(re);
	} else if (im <= 2e+119) {
		tmp = pow(re, -2.0);
	} else {
		tmp = re * (0.5 * fma(im, im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12400000.0)
		tmp = sin(re);
	elseif (im <= 2e+119)
		tmp = re ^ -2.0;
	else
		tmp = Float64(re * Float64(0.5 * fma(im, im, 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12400000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2e+119], N[Power[re, -2.0], $MachinePrecision], N[(re * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.24e7

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.9%

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

   if 1.24e7 < im < 1.99999999999999989e119

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 13.4%

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

   7. Taylor expanded in re around 0 13.3%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]

   9. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{{re}^{-2}} \]

   if 1.99999999999999989e119 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in re around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*r* 64.3%

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

      3. +-commutative 64.3%

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

      4. unpow2 64.3%

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

      5. fma-udef 64.3%

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

   10. Simplified 64.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12400000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12400000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2} + 0.08333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12400000.0) (sin re) (+ (pow re -2.0) 0.08333333333333333)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12400000.0) {
		tmp = sin(re);
	} else {
		tmp = pow(re, -2.0) + 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 <= 12400000.0d0) then
        tmp = sin(re)
    else
        tmp = (re ** (-2.0d0)) + 0.08333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12400000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.pow(re, -2.0) + 0.08333333333333333;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12400000.0:
		tmp = math.sin(re)
	else:
		tmp = math.pow(re, -2.0) + 0.08333333333333333
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12400000.0)
		tmp = sin(re);
	else
		tmp = Float64((re ^ -2.0) + 0.08333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12400000.0)
		tmp = sin(re);
	else
		tmp = (re ^ -2.0) + 0.08333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12400000.0], N[Sin[re], $MachinePrecision], N[(N[Power[re, -2.0], $MachinePrecision] + 0.08333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.24e7

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.9%

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

   if 1.24e7 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.4%

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

   7. Taylor expanded in re around 0 15.4%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 15.4%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 15.4%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 15.4%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 15.4%

       \[\leadsto 0.08333333333333333 + \color{blue}{{re}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12400000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2} + 0.08333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12400000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12400000.0) (sin re) (pow re -2.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12400000.0) {
		tmp = sin(re);
	} else {
		tmp = pow(re, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 12400000.0d0) then
        tmp = sin(re)
    else
        tmp = re ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12400000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12400000.0:
		tmp = math.sin(re)
	else:
		tmp = math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12400000.0)
		tmp = sin(re);
	else
		tmp = re ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12400000.0)
		tmp = sin(re);
	else
		tmp = re ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12400000.0], N[Sin[re], $MachinePrecision], N[Power[re, -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.24e7

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 69.9%

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

   if 1.24e7 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.4%

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

   7. Taylor expanded in re around 0 15.3%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]

   9. Applied egg-rr 15.3%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12400000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+25}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+159} \lor \neg \left(im \leq 1.05 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{\frac{3.3489797668038406 \cdot 10^{-7} - re \cdot re}{re + 0.0005787037037037037}}{0.006944444444444444 + re \cdot 0.9791666666666666}\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e+25)
   (sin re)
   (if (or (<= im 2.35e+159) (not (<= im 1.05e+196)))
     (/
      (/ (- 3.3489797668038406e-7 (* re re)) (+ re 0.0005787037037037037))
      (+ 0.006944444444444444 (* re 0.9791666666666666)))
     (+ 0.08333333333333333 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e+25) {
		tmp = sin(re);
	} else if ((im <= 2.35e+159) || !(im <= 1.05e+196)) {
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666));
	} else {
		tmp = 0.08333333333333333 + (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 <= 2.6d+25) then
        tmp = sin(re)
    else if ((im <= 2.35d+159) .or. (.not. (im <= 1.05d+196))) then
        tmp = ((3.3489797668038406d-7 - (re * re)) / (re + 0.0005787037037037037d0)) / (0.006944444444444444d0 + (re * 0.9791666666666666d0))
    else
        tmp = 0.08333333333333333d0 + (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6e+25) {
		tmp = Math.sin(re);
	} else if ((im <= 2.35e+159) || !(im <= 1.05e+196)) {
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666));
	} else {
		tmp = 0.08333333333333333 + (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e+25:
		tmp = math.sin(re)
	elif (im <= 2.35e+159) or not (im <= 1.05e+196):
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666))
	else:
		tmp = 0.08333333333333333 + (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e+25)
		tmp = sin(re);
	elseif ((im <= 2.35e+159) || !(im <= 1.05e+196))
		tmp = Float64(Float64(Float64(3.3489797668038406e-7 - Float64(re * re)) / Float64(re + 0.0005787037037037037)) / Float64(0.006944444444444444 + Float64(re * 0.9791666666666666)));
	else
		tmp = Float64(0.08333333333333333 + Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6e+25)
		tmp = sin(re);
	elseif ((im <= 2.35e+159) || ~((im <= 1.05e+196)))
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666));
	else
		tmp = 0.08333333333333333 + (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e+25], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 2.35e+159], N[Not[LessEqual[im, 1.05e+196]], $MachinePrecision]], N[(N[(N[(3.3489797668038406e-7 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(re + 0.0005787037037037037), $MachinePrecision]), $MachinePrecision] / N[(0.006944444444444444 + N[(re * 0.9791666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(re * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5999999999999998e25

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 68.8%

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

   if 2.5999999999999998e25 < im < 2.3500000000000002e159 or 1.05000000000000007e196 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.0%

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

   7. Taylor expanded in re around 0 15.0%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 15.0%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 15.0%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 15.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 2.0%

       \[\leadsto \color{blue}{\frac{0.0005787037037037037 - re}{0.006944444444444444 + \left(re + re \cdot -0.020833333333333332\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 2.0%

          \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \left(re + \color{blue}{-0.020833333333333332 \cdot re}\right)} \]

       2. distribute-rgt1-in 2.0%

          \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \color{blue}{\left(-0.020833333333333332 + 1\right) \cdot re}} \]

       3. metadata-eval 2.0%

          \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \color{blue}{0.9791666666666666} \cdot re} \]

   13. Simplified 2.0%

       \[\leadsto \color{blue}{\frac{0.0005787037037037037 - re}{0.006944444444444444 + 0.9791666666666666 \cdot re}} \]
   14. Step-by-step derivation

       1. sub-neg 2.0%

          \[\leadsto \frac{\color{blue}{0.0005787037037037037 + \left(-re\right)}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

       2. flip-+ 20.0%

          \[\leadsto \frac{\color{blue}{\frac{0.0005787037037037037 \cdot 0.0005787037037037037 - \left(-re\right) \cdot \left(-re\right)}{0.0005787037037037037 - \left(-re\right)}}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

       3. metadata-eval 20.0%

          \[\leadsto \frac{\frac{\color{blue}{3.3489797668038406 \cdot 10^{-7}} - \left(-re\right) \cdot \left(-re\right)}{0.0005787037037037037 - \left(-re\right)}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

   15. Applied egg-rr 20.0%

       \[\leadsto \frac{\color{blue}{\frac{3.3489797668038406 \cdot 10^{-7} - \left(-re\right) \cdot \left(-re\right)}{0.0005787037037037037 - \left(-re\right)}}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

   if 2.3500000000000002e159 < im < 1.05000000000000007e196

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 21.9%

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

   7. Taylor expanded in re around 0 21.8%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 21.8%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 21.8%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 21.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 31.0%

       \[\leadsto 0.08333333333333333 + \color{blue}{re \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+25}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+159} \lor \neg \left(im \leq 1.05 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{\frac{3.3489797668038406 \cdot 10^{-7} - re \cdot re}{re + 0.0005787037037037037}}{0.006944444444444444 + re \cdot 0.9791666666666666}\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.2% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.17:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + re \cdot 1.0208333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.17)
   re
   (/
    (/
     (* (+ re 0.0005787037037037037) (+ re 0.006944444444444444))
     (+ re -0.020833333333333332))
    (+ 0.006944444444444444 (* re 1.0208333333333333)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.17) {
		tmp = re;
	} else {
		tmp = (((re + 0.0005787037037037037) * (re + 0.006944444444444444)) / (re + -0.020833333333333332)) / (0.006944444444444444 + (re * 1.0208333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.17d0) then
        tmp = re
    else
        tmp = (((re + 0.0005787037037037037d0) * (re + 0.006944444444444444d0)) / (re + (-0.020833333333333332d0))) / (0.006944444444444444d0 + (re * 1.0208333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.17) {
		tmp = re;
	} else {
		tmp = (((re + 0.0005787037037037037) * (re + 0.006944444444444444)) / (re + -0.020833333333333332)) / (0.006944444444444444 + (re * 1.0208333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.17:
		tmp = re
	else:
		tmp = (((re + 0.0005787037037037037) * (re + 0.006944444444444444)) / (re + -0.020833333333333332)) / (0.006944444444444444 + (re * 1.0208333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.17)
		tmp = re;
	else
		tmp = Float64(Float64(Float64(Float64(re + 0.0005787037037037037) * Float64(re + 0.006944444444444444)) / Float64(re + -0.020833333333333332)) / Float64(0.006944444444444444 + Float64(re * 1.0208333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.17)
		tmp = re;
	else
		tmp = (((re + 0.0005787037037037037) * (re + 0.006944444444444444)) / (re + -0.020833333333333332)) / (0.006944444444444444 + (re * 1.0208333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.17], re, N[(N[(N[(N[(re + 0.0005787037037037037), $MachinePrecision] * N[(re + 0.006944444444444444), $MachinePrecision]), $MachinePrecision] / N[(re + -0.020833333333333332), $MachinePrecision]), $MachinePrecision] / N[(0.006944444444444444 + N[(re * 1.0208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.170000000000000012

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in re around 0 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*r* 56.9%

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

      3. +-commutative 56.9%

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

      4. unpow2 56.9%

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

      5. fma-udef 56.9%

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

   10. Simplified 56.9%

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

   11. Taylor expanded in im around 0 34.7%

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

   if 0.170000000000000012 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in re around 0 5.7%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 5.7%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 5.7%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 5.7%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 3.2%

       \[\leadsto \color{blue}{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(0.006944444444444444 + re\right)}{\left(\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 3.2%

          \[\leadsto \frac{\color{blue}{\left(0.006944444444444444 + re\right) \cdot \left(re + 0.0005787037037037037\right)}}{\left(\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)} \]

       2. associate-/l/ 16.6%

          \[\leadsto \color{blue}{\frac{\frac{\left(0.006944444444444444 + re\right) \cdot \left(re + 0.0005787037037037037\right)}{re + -0.020833333333333332}}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332}} \]

       3. *-commutative 16.6%

          \[\leadsto \frac{\frac{\color{blue}{\left(re + 0.0005787037037037037\right) \cdot \left(0.006944444444444444 + re\right)}}{re + -0.020833333333333332}}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \]

       4. +-commutative 16.6%

          \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \color{blue}{\left(re + 0.006944444444444444\right)}}{re + -0.020833333333333332}}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \]

       5. associate--l+ 16.6%

          \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{\color{blue}{0.006944444444444444 + \left(re - re \cdot -0.020833333333333332\right)}} \]

       6. *-commutative 16.6%

          \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + \left(re - \color{blue}{-0.020833333333333332 \cdot re}\right)} \]

       7. cancel-sign-sub-inv 16.6%

          \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + \color{blue}{\left(re + \left(--0.020833333333333332\right) \cdot re\right)}} \]

       8. distribute-rgt1-in 16.6%

          \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + \color{blue}{\left(\left(--0.020833333333333332\right) + 1\right) \cdot re}} \]

       9. metadata-eval 16.6%

          \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + \left(\color{blue}{0.020833333333333332} + 1\right) \cdot re} \]

       10. metadata-eval 16.6%

           \[\leadsto \frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + \color{blue}{1.0208333333333333} \cdot re} \]

   13. Simplified 16.6%

       \[\leadsto \color{blue}{\frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + 1.0208333333333333 \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.17:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.006944444444444444\right)}{re + -0.020833333333333332}}{0.006944444444444444 + re \cdot 1.0208333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.7% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{3.3489797668038406 \cdot 10^{-7} - re \cdot re}{re + 0.0005787037037037037}}{0.006944444444444444 + re \cdot 0.9791666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.1)
   re
   (/
    (/ (- 3.3489797668038406e-7 (* re re)) (+ re 0.0005787037037037037))
    (+ 0.006944444444444444 (* re 0.9791666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.1) {
		tmp = re;
	} else {
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.1d0) then
        tmp = re
    else
        tmp = ((3.3489797668038406d-7 - (re * re)) / (re + 0.0005787037037037037d0)) / (0.006944444444444444d0 + (re * 0.9791666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.1) {
		tmp = re;
	} else {
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.1:
		tmp = re
	else:
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.1)
		tmp = re;
	else
		tmp = Float64(Float64(Float64(3.3489797668038406e-7 - Float64(re * re)) / Float64(re + 0.0005787037037037037)) / Float64(0.006944444444444444 + Float64(re * 0.9791666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.1)
		tmp = re;
	else
		tmp = ((3.3489797668038406e-7 - (re * re)) / (re + 0.0005787037037037037)) / (0.006944444444444444 + (re * 0.9791666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.1], re, N[(N[(N[(3.3489797668038406e-7 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(re + 0.0005787037037037037), $MachinePrecision]), $MachinePrecision] / N[(0.006944444444444444 + N[(re * 0.9791666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.10000000000000009

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in re around 0 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*r* 56.9%

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

      3. +-commutative 56.9%

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

      4. unpow2 56.9%

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

      5. fma-udef 56.9%

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

   10. Simplified 56.9%

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

   11. Taylor expanded in im around 0 34.7%

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

   if 3.10000000000000009 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in re around 0 5.7%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 5.7%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 5.7%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 5.7%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 5.8%

       \[\leadsto \color{blue}{\frac{0.0005787037037037037 - re}{0.006944444444444444 + \left(re + re \cdot -0.020833333333333332\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 5.8%

          \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \left(re + \color{blue}{-0.020833333333333332 \cdot re}\right)} \]

       2. distribute-rgt1-in 5.8%

          \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \color{blue}{\left(-0.020833333333333332 + 1\right) \cdot re}} \]

       3. metadata-eval 5.8%

          \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \color{blue}{0.9791666666666666} \cdot re} \]

   13. Simplified 5.8%

       \[\leadsto \color{blue}{\frac{0.0005787037037037037 - re}{0.006944444444444444 + 0.9791666666666666 \cdot re}} \]
   14. Step-by-step derivation

       1. sub-neg 5.8%

          \[\leadsto \frac{\color{blue}{0.0005787037037037037 + \left(-re\right)}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

       2. flip-+ 21.1%

          \[\leadsto \frac{\color{blue}{\frac{0.0005787037037037037 \cdot 0.0005787037037037037 - \left(-re\right) \cdot \left(-re\right)}{0.0005787037037037037 - \left(-re\right)}}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

       3. metadata-eval 21.1%

          \[\leadsto \frac{\frac{\color{blue}{3.3489797668038406 \cdot 10^{-7}} - \left(-re\right) \cdot \left(-re\right)}{0.0005787037037037037 - \left(-re\right)}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]

   15. Applied egg-rr 21.1%

       \[\leadsto \frac{\color{blue}{\frac{3.3489797668038406 \cdot 10^{-7} - \left(-re\right) \cdot \left(-re\right)}{0.0005787037037037037 - \left(-re\right)}}}{0.006944444444444444 + 0.9791666666666666 \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{3.3489797668038406 \cdot 10^{-7} - re \cdot re}{re + 0.0005787037037037037}}{0.006944444444444444 + re \cdot 0.9791666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.9% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 58000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 58000.0) re (+ 0.08333333333333333 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 58000.0) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + (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 <= 58000.0d0) then
        tmp = re
    else
        tmp = 0.08333333333333333d0 + (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 58000.0) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 58000.0:
		tmp = re
	else:
		tmp = 0.08333333333333333 + (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 58000.0)
		tmp = re;
	else
		tmp = Float64(0.08333333333333333 + Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 58000.0)
		tmp = re;
	else
		tmp = 0.08333333333333333 + (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 58000.0], re, N[(0.08333333333333333 + N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 58000

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in re around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-*r* 48.6%

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

      3. +-commutative 48.6%

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

      4. unpow2 48.6%

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

      5. fma-udef 48.6%

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

   10. Simplified 48.6%

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

   11. Taylor expanded in im around 0 36.8%

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

   if 58000 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.2%

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

   7. Taylor expanded in re around 0 15.2%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 15.2%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 15.2%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 15.2%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 15.1%

       \[\leadsto 0.08333333333333333 + \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 58000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.9% accurate, 38.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 58000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 58000.0) re (* re re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 58000.0) {
		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 <= 58000.0d0) 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 <= 58000.0) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 58000.0:
		tmp = re
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 58000.0)
		tmp = re;
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 58000.0)
		tmp = re;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 58000.0], re, N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 58000

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in re around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-*r* 48.6%

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

      3. +-commutative 48.6%

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

      4. unpow2 48.6%

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

      5. fma-udef 48.6%

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

   10. Simplified 48.6%

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

   11. Taylor expanded in im around 0 36.8%

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

   if 58000 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.2%

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

   7. Taylor expanded in re around 0 15.1%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]

   9. Applied egg-rr 15.0%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 58000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.2% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{-5}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.9596001665972511\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 3.9e-5) re 0.9596001665972511))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.9e-5) {
		tmp = re;
	} else {
		tmp = 0.9596001665972511;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.9d-5) then
        tmp = re
    else
        tmp = 0.9596001665972511d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.9e-5) {
		tmp = re;
	} else {
		tmp = 0.9596001665972511;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.9e-5:
		tmp = re
	else:
		tmp = 0.9596001665972511
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.9e-5)
		tmp = re;
	else
		tmp = 0.9596001665972511;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.9e-5)
		tmp = re;
	else
		tmp = 0.9596001665972511;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.9e-5], re, 0.9596001665972511]

Derivation

1. Split input into 2 regimes

2. if re < 3.8999999999999999e-5

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in re around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r* 57.2%

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

      3. +-commutative 57.2%

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

      4. unpow2 57.2%

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

      5. fma-udef 57.2%

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

   10. Simplified 57.2%

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

   11. Taylor expanded in im around 0 34.8%

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

   if 3.8999999999999999e-5 < re

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 5.8%

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

   7. Taylor expanded in re around 0 5.7%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 5.7%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 5.7%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   9. Simplified 5.7%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

   11. Applied egg-rr 3.3%

       \[\leadsto \color{blue}{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.0005787037037037037\right)}{\left(\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332\right) \cdot \left(\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332\right)}} \]
   12. Step-by-step derivation

       1. times-frac 7.1%

          \[\leadsto \color{blue}{\frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \cdot \frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332}} \]

       2. +-commutative 7.1%

          \[\leadsto \frac{re + 0.0005787037037037037}{\color{blue}{\left(re + 0.006944444444444444\right)} - re \cdot -0.020833333333333332} \cdot \frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \]

       3. associate-+r- 7.1%

          \[\leadsto \frac{re + 0.0005787037037037037}{\color{blue}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)}} \cdot \frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \]

       4. +-commutative 7.1%

          \[\leadsto \frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)} \cdot \frac{re + 0.0005787037037037037}{\color{blue}{\left(re + 0.006944444444444444\right)} - re \cdot -0.020833333333333332} \]

       5. associate-+r- 7.1%

          \[\leadsto \frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)} \cdot \frac{re + 0.0005787037037037037}{\color{blue}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)}} \]

   13. Simplified 7.1%

       \[\leadsto \color{blue}{\frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)} \cdot \frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)}} \]

   14. Taylor expanded in re around inf 7.1%

       \[\leadsto \color{blue}{0.9596001665972511} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{-5}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.9596001665972511\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 4.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1.0212765957446808 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0212765957446808)

C

double code(double re, double im) {
	return -1.0212765957446808;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.0212765957446808d0
end function

Java

public static double code(double re, double im) {
	return -1.0212765957446808;
}

Python

def code(re, im):
	return -1.0212765957446808

Julia

function code(re, im)
	return -1.0212765957446808
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0212765957446808;
end

Wolfram

code[re_, im_] := -1.0212765957446808

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 9.6%

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

7. Taylor expanded in re around 0 9.5%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
8. Step-by-step derivation

   1. associate-*r/ 9.5%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 9.5%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

9. Simplified 9.5%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

11. Applied egg-rr 4.6%

    \[\leadsto \color{blue}{\frac{0.0005787037037037037 - re}{0.006944444444444444 + \left(re + re \cdot -0.020833333333333332\right)}} \]
12. Step-by-step derivation

    1. *-commutative 4.6%

       \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \left(re + \color{blue}{-0.020833333333333332 \cdot re}\right)} \]

    2. distribute-rgt1-in 4.6%

       \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \color{blue}{\left(-0.020833333333333332 + 1\right) \cdot re}} \]

    3. metadata-eval 4.6%

       \[\leadsto \frac{0.0005787037037037037 - re}{0.006944444444444444 + \color{blue}{0.9791666666666666} \cdot re} \]

13. Simplified 4.6%

    \[\leadsto \color{blue}{\frac{0.0005787037037037037 - re}{0.006944444444444444 + 0.9791666666666666 \cdot re}} \]

14. Taylor expanded in re around inf 4.6%

    \[\leadsto \color{blue}{-1.0212765957446808} \]

15. Final simplification 4.6%

    \[\leadsto -1.0212765957446808 \]
16. Add Preprocessing

Alternative 15: 4.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 9.6%

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

7. Taylor expanded in re around 0 9.5%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
8. Step-by-step derivation

   1. associate-*r/ 9.5%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 9.5%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

9. Simplified 9.5%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

10. Taylor expanded in re around inf 4.1%

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

11. Final simplification 4.1%

    \[\leadsto 0.08333333333333333 \]
12. Add Preprocessing

Alternative 16: 4.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.9596001665972511 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.9596001665972511)

C

double code(double re, double im) {
	return 0.9596001665972511;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.9596001665972511d0
end function

Java

public static double code(double re, double im) {
	return 0.9596001665972511;
}

Python

def code(re, im):
	return 0.9596001665972511

Julia

function code(re, im)
	return 0.9596001665972511
end

MATLAB

function tmp = code(re, im)
	tmp = 0.9596001665972511;
end

Wolfram

code[re_, im_] := 0.9596001665972511

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 9.6%

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

7. Taylor expanded in re around 0 9.5%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
8. Step-by-step derivation

   1. associate-*r/ 9.5%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 9.5%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

9. Simplified 9.5%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{{re}^{2}}} \]

11. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{\frac{\left(re + 0.0005787037037037037\right) \cdot \left(re + 0.0005787037037037037\right)}{\left(\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332\right) \cdot \left(\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332\right)}} \]
12. Step-by-step derivation

    1. times-frac 4.8%

       \[\leadsto \color{blue}{\frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \cdot \frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332}} \]

    2. +-commutative 4.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{\color{blue}{\left(re + 0.006944444444444444\right)} - re \cdot -0.020833333333333332} \cdot \frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \]

    3. associate-+r- 4.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{\color{blue}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)}} \cdot \frac{re + 0.0005787037037037037}{\left(0.006944444444444444 + re\right) - re \cdot -0.020833333333333332} \]

    4. +-commutative 4.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)} \cdot \frac{re + 0.0005787037037037037}{\color{blue}{\left(re + 0.006944444444444444\right)} - re \cdot -0.020833333333333332} \]

    5. associate-+r- 4.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)} \cdot \frac{re + 0.0005787037037037037}{\color{blue}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)}} \]

13. Simplified 4.8%

    \[\leadsto \color{blue}{\frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)} \cdot \frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - re \cdot -0.020833333333333332\right)}} \]

14. Taylor expanded in re around inf 4.8%

    \[\leadsto \color{blue}{0.9596001665972511} \]

15. Final simplification 4.8%

    \[\leadsto 0.9596001665972511 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))