math.sin on complex, real part

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

Alternatives: 11

Speedup: 1.0×

• 50.5% 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: 99.9% 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: 99.9% 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 99.6%

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

   1. distribute-lft-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-neg-out 99.6%

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

   4. sin-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. sin-neg 99.6%

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

   7. distribute-rgt-neg-out 99.6%

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

   8. neg-mul-1 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   9. associate-*r* 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   10. *-commutative 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

   11. distribute-lft-out-- 99.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   12. sub-neg 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   13. neg-sub0 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq 0.0295:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+117}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( t_0 \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (+ 1.0 (* im (* 0.5 im))))))
   (if (<= im 0.0295)
     t_0
     (if (<= im 3.3e+117)
       (* re (cosh im))
       (cast (! :precision binary32 (cast (! :precision binary64 t_0))))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (1.0 + (im * (0.5 * im)));
	double tmp;
	if (im <= 0.0295) {
		tmp = t_0;
	} else if (im <= 3.3e+117) {
		tmp = re * cosh(im);
	} else {
		double tmp_3 = t_0;
		double tmp_2 = (float) tmp_3;
		tmp = (double) tmp_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    if (im <= 0.0295d0) then
        tmp = t_0
    else if (im <= 3.3d+117) then
        tmp = re * cosh(im)
    else
        tmp_3 = t_0
        tmp_2 = real(tmp_3, 4)
        tmp = real(tmp_2, 8)
    end if
    code = tmp
end function

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))))
	tmp = 0.0
	if (im <= 0.0295)
		tmp = t_0;
	elseif (im <= 3.3e+117)
		tmp = Float64(re * cosh(im));
	else
		tmp_3 = t_0
		tmp_2 = Float32(tmp_3)
		tmp = Float64(tmp_2);
	end
	return tmp
end

MATLAB

function tmp_5 = code(re, im)
	t_0 = sin(re) * (1.0 + (im * (0.5 * im)));
	tmp = 0.0;
	if (im <= 0.0295)
		tmp = t_0;
	elseif (im <= 3.3e+117)
		tmp = re * cosh(im);
	else
		tmp_4 = t_0;
		tmp_3 = single(tmp_4);
		tmp = double(tmp_3);
	end
	tmp_5 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if im < 0.029499999999999998

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. sin-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. sin-neg 99.5%

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

      7. distribute-rgt-neg-out 99.5%

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

      8. neg-mul-1 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.5%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 84.6%

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

      1. *-lft-identity 84.6%

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

      2. associate-*r* 84.6%

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

      3. distribute-rgt-out 84.6%

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

      4. unpow2 84.6%

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

      5. associate-*r* 84.6%

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

   6. Simplified 84.6%

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

   if 0.029499999999999998 < im < 3.2999999999999998e117

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 88.2%

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

      1. flip3-+ 5.8%

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

      2. associate-*r/ 5.8%

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

      3. associate-/l* 5.8%

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

      4. *-commutative 5.8%

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

      5. clear-num 5.8%

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

      6. flip3-+ 88.2%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{-im} + e^{im}}}} \]

      7. +-commutative 88.2%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{im} + e^{-im}}}} \]

      8. cosh-undef 88.2%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{2 \cdot \cosh im}}} \]

   6. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{\frac{re \cdot 0.5}{\frac{1}{2 \cdot \cosh im}}} \]
   7. Step-by-step derivation

      1. associate-/l* 88.2%

         \[\leadsto \color{blue}{\frac{re}{\frac{\frac{1}{2 \cdot \cosh im}}{0.5}}} \]

      2. associate-/r* 88.2%

         \[\leadsto \frac{re}{\frac{\color{blue}{\frac{\frac{1}{2}}{\cosh im}}}{0.5}} \]

      3. metadata-eval 88.2%

         \[\leadsto \frac{re}{\frac{\frac{\color{blue}{0.5}}{\cosh im}}{0.5}} \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{re}{\frac{\frac{0.5}{\cosh im}}{0.5}}} \]
   9. Step-by-step derivation

      1. clear-num 88.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5}{\cosh im}}{0.5}}{re}}} \]

      2. associate-/r/ 88.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{\cosh im}}{0.5}} \cdot re} \]

      3. clear-num 88.2%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\cosh im}}} \cdot re \]

      4. associate-/r/ 88.2%

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

      5. metadata-eval 88.2%

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

      6. *-lft-identity 88.2%

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

   10. Applied egg-rr 88.2%

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

   if 3.2999999999999998e117 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.1%

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

      1. *-lft-identity 85.1%

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

      2. associate-*r* 85.1%

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

      3. distribute-rgt-out 85.1%

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

      4. unpow2 85.1%

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

      5. associate-*r* 85.1%

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

   6. Simplified 85.1%

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

      1. rewrite-binary64/binary32-simplify 100.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot im\right) \cdot im\right)} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 100.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot im\right) \cdot im\right)} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0295:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+117}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( \sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \]

Alternative 3: 84.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.082:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.082)
   (* (sin re) (+ 1.0 (* im (* 0.5 im))))
   (if (<= im 1.35e+154) (* re (cosh im)) (* (sin re) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.082) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 1.35e+154) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.082d0) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else if (im <= 1.35d+154) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.082) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 1.35e+154) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.082:
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	elif im <= 1.35e+154:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.082)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 1.35e+154)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.082)
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	elseif (im <= 1.35e+154)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.082], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0820000000000000034

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. sin-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. sin-neg 99.5%

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

      7. distribute-rgt-neg-out 99.5%

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

      8. neg-mul-1 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.5%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 84.6%

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

      1. *-lft-identity 84.6%

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

      2. associate-*r* 84.6%

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

      3. distribute-rgt-out 84.6%

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

      4. unpow2 84.6%

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

      5. associate-*r* 84.6%

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

   6. Simplified 84.6%

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

   if 0.0820000000000000034 < 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-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.0%

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

      1. flip3-+ 4.1%

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

      2. associate-*r/ 4.1%

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

      3. associate-/l* 4.1%

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

      4. *-commutative 4.1%

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

      5. clear-num 4.1%

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

      6. flip3-+ 75.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{-im} + e^{im}}}} \]

      7. +-commutative 75.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{im} + e^{-im}}}} \]

      8. cosh-undef 75.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{2 \cdot \cosh im}}} \]

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{re \cdot 0.5}{\frac{1}{2 \cdot \cosh im}}} \]
   7. Step-by-step derivation

      1. associate-/l* 75.0%

         \[\leadsto \color{blue}{\frac{re}{\frac{\frac{1}{2 \cdot \cosh im}}{0.5}}} \]

      2. associate-/r* 75.0%

         \[\leadsto \frac{re}{\frac{\color{blue}{\frac{\frac{1}{2}}{\cosh im}}}{0.5}} \]

      3. metadata-eval 75.0%

         \[\leadsto \frac{re}{\frac{\frac{\color{blue}{0.5}}{\cosh im}}{0.5}} \]

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{re}{\frac{\frac{0.5}{\cosh im}}{0.5}}} \]
   9. Step-by-step derivation

      1. clear-num 74.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5}{\cosh im}}{0.5}}{re}}} \]

      2. associate-/r/ 75.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{\cosh im}}{0.5}} \cdot re} \]

      3. clear-num 75.0%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\cosh im}}} \cdot re \]

      4. associate-/r/ 75.0%

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

      5. metadata-eval 75.0%

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

      6. *-lft-identity 75.0%

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

   10. Applied egg-rr 75.0%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{{im}^{2} \cdot \sin re} \]
   10. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{\sin re \cdot {im}^{2}} \]

       2. unpow2 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.082:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 4: 71.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0106:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0106)
   (sin re)
   (if (<= im 1.35e+154) (* re (cosh im)) (* (sin re) (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0106) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0106d0) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0106) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0106:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0106)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0106)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0106], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0106

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. sin-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. sin-neg 99.5%

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

      7. distribute-rgt-neg-out 99.5%

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

      8. neg-mul-1 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.5%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 66.3%

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

   if 0.0106 < 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-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.0%

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

      1. flip3-+ 4.1%

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

      2. associate-*r/ 4.1%

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

      3. associate-/l* 4.1%

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

      4. *-commutative 4.1%

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

      5. clear-num 4.1%

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

      6. flip3-+ 75.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{-im} + e^{im}}}} \]

      7. +-commutative 75.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{im} + e^{-im}}}} \]

      8. cosh-undef 75.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{2 \cdot \cosh im}}} \]

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{re \cdot 0.5}{\frac{1}{2 \cdot \cosh im}}} \]
   7. Step-by-step derivation

      1. associate-/l* 75.0%

         \[\leadsto \color{blue}{\frac{re}{\frac{\frac{1}{2 \cdot \cosh im}}{0.5}}} \]

      2. associate-/r* 75.0%

         \[\leadsto \frac{re}{\frac{\color{blue}{\frac{\frac{1}{2}}{\cosh im}}}{0.5}} \]

      3. metadata-eval 75.0%

         \[\leadsto \frac{re}{\frac{\frac{\color{blue}{0.5}}{\cosh im}}{0.5}} \]

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{re}{\frac{\frac{0.5}{\cosh im}}{0.5}}} \]
   9. Step-by-step derivation

      1. clear-num 74.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5}{\cosh im}}{0.5}}{re}}} \]

      2. associate-/r/ 75.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{\cosh im}}{0.5}} \cdot re} \]

      3. clear-num 75.0%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\cosh im}}} \cdot re \]

      4. associate-/r/ 75.0%

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

      5. metadata-eval 75.0%

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

      6. *-lft-identity 75.0%

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

   10. Applied egg-rr 75.0%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{{im}^{2} \cdot \sin re} \]
   10. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{\sin re \cdot {im}^{2}} \]

       2. unpow2 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0106:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 5: 67.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0175:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0175) (sin re) (* re (cosh im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0175) {
		tmp = sin(re);
	} else {
		tmp = re * cosh(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 <= 0.0175d0) then
        tmp = sin(re)
    else
        tmp = re * cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0175) {
		tmp = Math.sin(re);
	} else {
		tmp = re * Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0175:
		tmp = math.sin(re)
	else:
		tmp = re * math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0175)
		tmp = sin(re);
	else
		tmp = Float64(re * cosh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0175)
		tmp = sin(re);
	else
		tmp = re * cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0175], N[Sin[re], $MachinePrecision], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.017500000000000002

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. sin-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. sin-neg 99.5%

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

      7. distribute-rgt-neg-out 99.5%

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

      8. neg-mul-1 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.5%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 66.3%

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

   if 0.017500000000000002 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 77.0%

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

      1. flip3-+ 1.6%

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

      2. associate-*r/ 1.6%

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

      3. associate-/l* 1.6%

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

      4. *-commutative 1.6%

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

      5. clear-num 1.6%

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

      6. flip3-+ 77.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{-im} + e^{im}}}} \]

      7. +-commutative 77.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{e^{im} + e^{-im}}}} \]

      8. cosh-undef 77.0%

         \[\leadsto \frac{re \cdot 0.5}{\frac{1}{\color{blue}{2 \cdot \cosh im}}} \]

   6. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\frac{re \cdot 0.5}{\frac{1}{2 \cdot \cosh im}}} \]
   7. Step-by-step derivation

      1. associate-/l* 77.0%

         \[\leadsto \color{blue}{\frac{re}{\frac{\frac{1}{2 \cdot \cosh im}}{0.5}}} \]

      2. associate-/r* 77.0%

         \[\leadsto \frac{re}{\frac{\color{blue}{\frac{\frac{1}{2}}{\cosh im}}}{0.5}} \]

      3. metadata-eval 77.0%

         \[\leadsto \frac{re}{\frac{\frac{\color{blue}{0.5}}{\cosh im}}{0.5}} \]

   8. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{re}{\frac{\frac{0.5}{\cosh im}}{0.5}}} \]
   9. Step-by-step derivation

      1. clear-num 77.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5}{\cosh im}}{0.5}}{re}}} \]

      2. associate-/r/ 77.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{\cosh im}}{0.5}} \cdot re} \]

      3. clear-num 77.0%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\cosh im}}} \cdot re \]

      4. associate-/r/ 77.0%

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

      5. metadata-eval 77.0%

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

      6. *-lft-identity 77.0%

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

   10. Applied egg-rr 77.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0175:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \]

Alternative 6: 60.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.8 \cdot 10^{+23}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+142}:\\ \;\;\;\;re + \left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(0.5 \cdot im\right)\right) \cdot \left(2 \cdot \left(re \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.8e+23)
   (sin re)
   (if (<= im 6e+142)
     (+ re (* (* re (* re re)) -0.16666666666666666))
     (* (+ 1.0 (* im (* 0.5 im))) (* 2.0 (* re 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.8e+23) {
		tmp = sin(re);
	} else if (im <= 6e+142) {
		tmp = re + ((re * (re * re)) * -0.16666666666666666);
	} else {
		tmp = (1.0 + (im * (0.5 * im))) * (2.0 * (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 <= 9.8d+23) then
        tmp = sin(re)
    else if (im <= 6d+142) then
        tmp = re + ((re * (re * re)) * (-0.16666666666666666d0))
    else
        tmp = (1.0d0 + (im * (0.5d0 * im))) * (2.0d0 * (re * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.8e+23) {
		tmp = Math.sin(re);
	} else if (im <= 6e+142) {
		tmp = re + ((re * (re * re)) * -0.16666666666666666);
	} else {
		tmp = (1.0 + (im * (0.5 * im))) * (2.0 * (re * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.8e+23:
		tmp = math.sin(re)
	elif im <= 6e+142:
		tmp = re + ((re * (re * re)) * -0.16666666666666666)
	else:
		tmp = (1.0 + (im * (0.5 * im))) * (2.0 * (re * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.8e+23)
		tmp = sin(re);
	elseif (im <= 6e+142)
		tmp = Float64(re + Float64(Float64(re * Float64(re * re)) * -0.16666666666666666));
	else
		tmp = Float64(Float64(1.0 + Float64(im * Float64(0.5 * im))) * Float64(2.0 * Float64(re * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.8e+23)
		tmp = sin(re);
	elseif (im <= 6e+142)
		tmp = re + ((re * (re * re)) * -0.16666666666666666);
	else
		tmp = (1.0 + (im * (0.5 * im))) * (2.0 * (re * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.8e+23], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6e+142], N[(re + N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.8000000000000006e23

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. sin-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. sin-neg 99.5%

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

      7. distribute-rgt-neg-out 99.5%

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

      8. neg-mul-1 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.5%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 66.0%

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

   if 9.8000000000000006e23 < im < 5.99999999999999949e142

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 2.5%

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

   5. Taylor expanded in re around 0 25.3%

      \[\leadsto \color{blue}{re + -0.16666666666666666 \cdot {re}^{3}} \]
   6. Step-by-step derivation

      1. *-commutative 25.3%

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

   7. Simplified 25.3%

      \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. unpow3 25.3%

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

   9. Applied egg-rr 25.3%

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

   if 5.99999999999999949e142 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 95.1%

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

      1. *-lft-identity 95.1%

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

      2. associate-*r* 95.1%

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

      3. distribute-rgt-out 95.1%

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

      4. unpow2 95.1%

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

      5. associate-*r* 95.1%

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

   6. Simplified 95.1%

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

   8. Applied egg-rr 95.1%

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

   10. Applied egg-rr 95.1%

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

   11. Taylor expanded in re around 0 74.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.8 \cdot 10^{+23}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+142}:\\ \;\;\;\;re + \left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(0.5 \cdot im\right)\right) \cdot \left(2 \cdot \left(re \cdot 2\right)\right)\\ \end{array} \]

Alternative 7: 48.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+60} \lor \neg \left(re \leq 1.8 \cdot 10^{+91}\right):\\ \;\;\;\;re + \left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -3.6e+60) (not (<= re 1.8e+91)))
   (+ re (* (* re (* re re)) -0.16666666666666666))
   (* re (+ 1.0 (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -3.6e+60) || !(re <= 1.8e+91)) {
		tmp = re + ((re * (re * re)) * -0.16666666666666666);
	} 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 ((re <= (-3.6d+60)) .or. (.not. (re <= 1.8d+91))) then
        tmp = re + ((re * (re * re)) * (-0.16666666666666666d0))
    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 ((re <= -3.6e+60) || !(re <= 1.8e+91)) {
		tmp = re + ((re * (re * re)) * -0.16666666666666666);
	} else {
		tmp = re * (1.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -3.6e+60) or not (re <= 1.8e+91):
		tmp = re + ((re * (re * re)) * -0.16666666666666666)
	else:
		tmp = re * (1.0 + (im * (0.5 * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -3.6e+60) || !(re <= 1.8e+91))
		tmp = Float64(re + Float64(Float64(re * Float64(re * re)) * -0.16666666666666666));
	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 ((re <= -3.6e+60) || ~((re <= 1.8e+91)))
		tmp = re + ((re * (re * re)) * -0.16666666666666666);
	else
		tmp = re * (1.0 + (im * (0.5 * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -3.6e+60], N[Not[LessEqual[re, 1.8e+91]], $MachinePrecision]], N[(re + N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $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 re < -3.59999999999999968e60 or 1.8e91 < 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-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 54.6%

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

   5. Taylor expanded in re around 0 27.9%

      \[\leadsto \color{blue}{re + -0.16666666666666666 \cdot {re}^{3}} \]
   6. Step-by-step derivation

      1. *-commutative 27.9%

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

   7. Simplified 27.9%

      \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. unpow3 27.9%

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

   9. Applied egg-rr 27.9%

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

   if -3.59999999999999968e60 < re < 1.8e91

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. cancel-sign-sub 99.4%

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

      3. distribute-rgt-neg-out 99.4%

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

      4. sin-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. sin-neg 99.4%

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

      7. distribute-rgt-neg-out 99.4%

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

      8. neg-mul-1 99.4%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.4%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.4%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.4%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.4%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in im around 0 79.0%

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

      1. *-lft-identity 79.0%

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

      2. associate-*r* 79.0%

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

      3. distribute-rgt-out 79.0%

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

      4. unpow2 79.0%

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

      5. associate-*r* 79.0%

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

   6. Simplified 79.0%

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

   7. Taylor expanded in re around 0 66.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+60} \lor \neg \left(re \leq 1.8 \cdot 10^{+91}\right):\\ \;\;\;\;re + \left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 47.7% 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 99.6%

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

   1. distribute-lft-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-neg-out 99.6%

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

   4. sin-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. sin-neg 99.6%

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

   7. distribute-rgt-neg-out 99.6%

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

   8. neg-mul-1 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   9. associate-*r* 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   10. *-commutative 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

   11. distribute-lft-out-- 99.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   12. sub-neg 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   13. neg-sub0 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in im around 0 79.3%

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

   1. *-lft-identity 79.3%

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

   2. associate-*r* 79.3%

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

   3. distribute-rgt-out 79.3%

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

   4. unpow2 79.3%

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

   5. associate-*r* 79.3%

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

6. Simplified 79.3%

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

7. Taylor expanded in re around 0 47.9%

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

8. Final simplification 47.9%

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

Alternative 9: 36.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0105:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.0105) re (* re (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0105) {
		tmp = re;
	} else {
		tmp = re * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0105d0) then
        tmp = re
    else
        tmp = re * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0105) {
		tmp = re;
	} else {
		tmp = re * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0105:
		tmp = re
	else:
		tmp = re * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0105)
		tmp = re;
	else
		tmp = Float64(re * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0105)
		tmp = re;
	else
		tmp = re * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0105], re, N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0105000000000000007

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. sin-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. sin-neg 99.5%

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

      7. distribute-rgt-neg-out 99.5%

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

      8. neg-mul-1 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 99.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 99.5%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 99.5%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 55.8%

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

   5. Taylor expanded in im around 0 32.9%

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

   if 0.0105000000000000007 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. sin-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. sin-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      9. associate-*r* 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      13. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.5%

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

      1. *-lft-identity 62.5%

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

      2. associate-*r* 62.5%

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

      3. distribute-rgt-out 62.5%

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

      4. unpow2 62.5%

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

      5. associate-*r* 62.5%

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

   6. Simplified 62.5%

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

   8. Applied egg-rr 61.7%

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

   9. Taylor expanded in re around 0 49.5%

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

   10. Taylor expanded in im around inf 49.5%

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

       1. *-commutative 49.5%

          \[\leadsto \color{blue}{re \cdot {im}^{2}} \]

       2. unpow2 49.5%

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

   12. Simplified 49.5%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0105:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.0)

C

double code(double re, double im) {
	return -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.0d0
end function

Java

public static double code(double re, double im) {
	return -3.0;
}

Python

def code(re, im):
	return -3.0

Julia

function code(re, im)
	return -3.0
end

MATLAB

function tmp = code(re, im)
	tmp = -3.0;
end

Wolfram

code[re_, im_] := -3.0

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-neg-out 99.6%

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

   4. sin-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. sin-neg 99.6%

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

   7. distribute-rgt-neg-out 99.6%

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

   8. neg-mul-1 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   9. associate-*r* 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   10. *-commutative 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

   11. distribute-lft-out-- 99.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   12. sub-neg 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   13. neg-sub0 99.6%

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

3. Simplified 99.6%

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

5. Applied egg-rr 3.9%

   \[\leadsto \color{blue}{-3 - \sin re} \]

6. Taylor expanded in re around 0 3.9%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 3.9%

   \[\leadsto -3 \]

Alternative 11: 26.2% 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 99.6%

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

   1. distribute-lft-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-neg-out 99.6%

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

   4. sin-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. sin-neg 99.6%

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

   7. distribute-rgt-neg-out 99.6%

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

   8. neg-mul-1 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   9. associate-*r* 99.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   10. *-commutative 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} - \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} \cdot -1\right)} \]

   11. distribute-lft-out-- 99.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   12. sub-neg 99.6%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   13. neg-sub0 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in re around 0 60.7%

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

5. Taylor expanded in im around 0 25.5%

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

6. Final simplification 25.5%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023297 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))