math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 9

Speedup: 1.0×

• 49.5% 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: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{if}\;im \leq 640:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+63}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (fma im im 2.0))))
   (if (<= im 640.0)
     t_0
     (if (<= im 3.4e+63)
       (pow (* (sin re) -2.0) -2.0)
       (if (<= im 1.35e+154)
         (+ re (* re (* 0.041666666666666664 (pow im 4.0))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * fma(im, im, 2.0);
	double tmp;
	if (im <= 640.0) {
		tmp = t_0;
	} else if (im <= 3.4e+63) {
		tmp = pow((sin(re) * -2.0), -2.0);
	} else if (im <= 1.35e+154) {
		tmp = re + (re * (0.041666666666666664 * pow(im, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0))
	tmp = 0.0
	if (im <= 640.0)
		tmp = t_0;
	elseif (im <= 3.4e+63)
		tmp = Float64(sin(re) * -2.0) ^ -2.0;
	elseif (im <= 1.35e+154)
		tmp = Float64(re + Float64(re * Float64(0.041666666666666664 * (im ^ 4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 640.0], t$95$0, If[LessEqual[im, 3.4e+63], N[Power[N[(N[Sin[re], $MachinePrecision] * -2.0), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re + N[(re * N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 640 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 640 < im < 3.3999999999999999e63

   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 34.1%

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

   if 3.3999999999999999e63 < 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

   5. Taylor expanded in re around 0 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in im around 0 76.9%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 76.9%

         \[\leadsto re + \color{blue}{\left(0.5 \cdot \left({im}^{2} \cdot re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)\right)} \]

      2. associate-*r* 76.9%

         \[\leadsto re + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} + 0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)\right) \]

      3. associate-*r* 76.9%

         \[\leadsto re + \left(\left(0.5 \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re}\right) \]

      4. distribute-rgt-out 76.9%

         \[\leadsto re + \color{blue}{re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   10. Simplified 76.9%

       \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   11. Taylor expanded in im around inf 76.9%

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

       1. associate-*r* 76.9%

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

       2. *-commutative 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 640:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+63}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.022 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.022) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (* (* 0.5 re) (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.022) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.022) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.022], 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[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.021999999999999999 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 89.1%

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

   7. Simplified 89.1%

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

   if 0.021999999999999999 < 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

   5. Taylor expanded in re around 0 82.8%

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

   7. Simplified 82.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.022 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1050:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+63}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1050.0)
   (sin re)
   (if (<= im 9.5e+63)
     (pow (* (sin re) -2.0) -2.0)
     (+ re (* re (* 0.041666666666666664 (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1050.0) {
		tmp = sin(re);
	} else if (im <= 9.5e+63) {
		tmp = pow((sin(re) * -2.0), -2.0);
	} else {
		tmp = re + (re * (0.041666666666666664 * pow(im, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1050.0d0) then
        tmp = sin(re)
    else if (im <= 9.5d+63) then
        tmp = (sin(re) * (-2.0d0)) ** (-2.0d0)
    else
        tmp = re + (re * (0.041666666666666664d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1050.0) {
		tmp = Math.sin(re);
	} else if (im <= 9.5e+63) {
		tmp = Math.pow((Math.sin(re) * -2.0), -2.0);
	} else {
		tmp = re + (re * (0.041666666666666664 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1050.0:
		tmp = math.sin(re)
	elif im <= 9.5e+63:
		tmp = math.pow((math.sin(re) * -2.0), -2.0)
	else:
		tmp = re + (re * (0.041666666666666664 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1050.0)
		tmp = sin(re);
	elseif (im <= 9.5e+63)
		tmp = Float64(sin(re) * -2.0) ^ -2.0;
	else
		tmp = Float64(re + Float64(re * Float64(0.041666666666666664 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1050.0)
		tmp = sin(re);
	elseif (im <= 9.5e+63)
		tmp = (sin(re) * -2.0) ^ -2.0;
	else
		tmp = re + (re * (0.041666666666666664 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1050.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 9.5e+63], N[Power[N[(N[Sin[re], $MachinePrecision] * -2.0), $MachinePrecision], -2.0], $MachinePrecision], N[(re + N[(re * N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1050

   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 72.6%

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

   if 1050 < im < 9.5000000000000003e63

   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 34.1%

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

   if 9.5000000000000003e63 < 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 re around 0 70.4%

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

   7. Simplified 70.4%

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

   8. Taylor expanded in im around 0 70.4%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 70.4%

         \[\leadsto re + \color{blue}{\left(0.5 \cdot \left({im}^{2} \cdot re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)\right)} \]

      2. associate-*r* 70.4%

         \[\leadsto re + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} + 0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)\right) \]

      3. associate-*r* 70.4%

         \[\leadsto re + \left(\left(0.5 \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re}\right) \]

      4. distribute-rgt-out 70.4%

         \[\leadsto re + \color{blue}{re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   10. Simplified 70.4%

       \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   11. Taylor expanded in im around inf 70.4%

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

       1. associate-*r* 70.4%

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

       2. *-commutative 70.4%

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

   13. Simplified 70.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1050:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+63}:\\ \;\;\;\;{\left(\sin re \cdot -2\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0)
   (sin re)
   (if (<= im 8.2e+61)
     (/ 0.25 (pow re 2.0))
     (+ re (* re (* 0.041666666666666664 (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = sin(re);
	} else if (im <= 8.2e+61) {
		tmp = 0.25 / pow(re, 2.0);
	} else {
		tmp = re + (re * (0.041666666666666664 * pow(im, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 720.0d0) then
        tmp = sin(re)
    else if (im <= 8.2d+61) then
        tmp = 0.25d0 / (re ** 2.0d0)
    else
        tmp = re + (re * (0.041666666666666664d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = Math.sin(re);
	} else if (im <= 8.2e+61) {
		tmp = 0.25 / Math.pow(re, 2.0);
	} else {
		tmp = re + (re * (0.041666666666666664 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = math.sin(re)
	elif im <= 8.2e+61:
		tmp = 0.25 / math.pow(re, 2.0)
	else:
		tmp = re + (re * (0.041666666666666664 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = sin(re);
	elseif (im <= 8.2e+61)
		tmp = Float64(0.25 / (re ^ 2.0));
	else
		tmp = Float64(re + Float64(re * Float64(0.041666666666666664 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = sin(re);
	elseif (im <= 8.2e+61)
		tmp = 0.25 / (re ^ 2.0);
	else
		tmp = re + (re * (0.041666666666666664 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 8.2e+61], N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $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 72.6%

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

   if 720 < im < 8.19999999999999944e61

   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 34.1%

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

   7. Taylor expanded in re around 0 34.2%

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

   if 8.19999999999999944e61 < 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 re around 0 70.4%

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

   7. Simplified 70.4%

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

   8. Taylor expanded in im around 0 70.4%

      \[\leadsto \color{blue}{re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot re\right) + 0.5 \cdot \left({im}^{2} \cdot re\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 70.4%

         \[\leadsto re + \color{blue}{\left(0.5 \cdot \left({im}^{2} \cdot re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)\right)} \]

      2. associate-*r* 70.4%

         \[\leadsto re + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} + 0.041666666666666664 \cdot \left({im}^{4} \cdot re\right)\right) \]

      3. associate-*r* 70.4%

         \[\leadsto re + \left(\left(0.5 \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot re}\right) \]

      4. distribute-rgt-out 70.4%

         \[\leadsto re + \color{blue}{re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   10. Simplified 70.4%

       \[\leadsto \color{blue}{re + re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   11. Taylor expanded in im around inf 70.4%

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

       1. associate-*r* 70.4%

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

       2. *-commutative 70.4%

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

   13. Simplified 70.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;re + re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 520.0)
   (sin re)
   (if (<= im 6.5e+98)
     (/ 0.25 (pow re 2.0))
     (* re (+ 1.0 (* im (* 0.5 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = sin(re);
	} else if (im <= 6.5e+98) {
		tmp = 0.25 / pow(re, 2.0);
	} else {
		tmp = re * (1.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 520.0d0) then
        tmp = sin(re)
    else if (im <= 6.5d+98) then
        tmp = 0.25d0 / (re ** 2.0d0)
    else
        tmp = re * (1.0d0 + (im * (0.5d0 * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.5e+98) {
		tmp = 0.25 / Math.pow(re, 2.0);
	} else {
		tmp = re * (1.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 520.0:
		tmp = math.sin(re)
	elif im <= 6.5e+98:
		tmp = 0.25 / math.pow(re, 2.0)
	else:
		tmp = re * (1.0 + (im * (0.5 * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 520.0)
		tmp = sin(re);
	elseif (im <= 6.5e+98)
		tmp = Float64(0.25 / (re ^ 2.0));
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 520.0)
		tmp = sin(re);
	elseif (im <= 6.5e+98)
		tmp = 0.25 / (re ^ 2.0);
	else
		tmp = re * (1.0 + (im * (0.5 * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 520.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.5e+98], N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 520

   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 72.6%

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

   if 520 < im < 6.4999999999999999e98

   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 21.6%

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

   7. Taylor expanded in re around 0 21.5%

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

   if 6.4999999999999999e98 < 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 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in re around 0 60.8%

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

      1. add-sqr-sqrt 60.8%

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

      2. pow2 60.8%

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

      3. *-commutative 60.8%

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

      4. sqrt-prod 60.8%

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

      5. unpow2 60.8%

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

      6. sqrt-prod 60.8%

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

      7. add-sqr-sqrt 60.8%

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

   10. Applied egg-rr 60.8%

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

       1. unpow2 60.8%

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

       2. *-commutative 60.8%

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

       3. *-commutative 60.8%

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

       4. swap-sqr 60.8%

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

       5. rem-square-sqrt 60.8%

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

       6. associate-*r* 60.8%

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

   12. Applied egg-rr 60.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.112:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.112) (sin re) (* re (+ 1.0 (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.112) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.112d0) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + (im * (0.5d0 * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.112) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.112:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + (im * (0.5 * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.112)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.112)
		tmp = sin(re);
	else
		tmp = re * (1.0 + (im * (0.5 * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.112], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.112000000000000002

   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 72.9%

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

   if 0.112000000000000002 < 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 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in re around 0 47.9%

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

      1. add-sqr-sqrt 47.9%

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

      2. pow2 47.9%

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

      3. *-commutative 47.9%

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

      4. sqrt-prod 47.9%

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

      5. unpow2 47.9%

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

      6. sqrt-prod 47.9%

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

      7. add-sqr-sqrt 47.9%

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

   10. Applied egg-rr 47.9%

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

       1. unpow2 47.9%

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

       2. *-commutative 47.9%

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

       3. *-commutative 47.9%

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

       4. swap-sqr 47.9%

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

       5. rem-square-sqrt 47.9%

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

       6. associate-*r* 47.9%

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

   12. Applied egg-rr 47.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.112:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.5% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-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 81.5%

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

7. Simplified 81.5%

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

8. Taylor expanded in re around 0 48.7%

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

   1. add-sqr-sqrt 48.7%

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

   2. pow2 48.7%

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

   3. *-commutative 48.7%

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

   4. sqrt-prod 48.7%

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

   5. unpow2 48.7%

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

   6. sqrt-prod 25.6%

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

   7. add-sqr-sqrt 48.7%

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

10. Applied egg-rr 48.7%

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

    1. unpow2 48.7%

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

    2. *-commutative 48.7%

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

    3. *-commutative 48.7%

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

    4. swap-sqr 48.7%

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

    5. rem-square-sqrt 48.7%

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

    6. associate-*r* 48.7%

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

12. Applied egg-rr 48.7%

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

13. Final simplification 48.7%

    \[\leadsto re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right) \]
14. Add Preprocessing

Alternative 9: 25.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. distribute-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 81.5%

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

7. Simplified 81.5%

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

8. Taylor expanded in re around 0 48.7%

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

9. Taylor expanded in im around 0 27.7%

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

10. Final simplification 27.7%

    \[\leadsto re \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))