math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 15

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 260.0)
   (cos re)
   (if (<= im 1.95e+93)
     (pow re -2.0)
     (if (<= im 1.3e+154) (log1p (expm1 re)) (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = cos(re);
	} else if (im <= 1.95e+93) {
		tmp = pow(re, -2.0);
	} else if (im <= 1.3e+154) {
		tmp = log1p(expm1(re));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 260.0)
		tmp = cos(re);
	elseif (im <= 1.95e+93)
		tmp = re ^ -2.0;
	elseif (im <= 1.3e+154)
		tmp = log1p(expm1(re));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 260.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.95e+93], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 1.3e+154], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 260

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   if 260 < im < 1.9500000000000001e93

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. unpow2 4.2%

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

      3. fma-define 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in im around inf 4.2%

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

      1. *-commutative 4.2%

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

      2. associate-*r* 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in re around 0 7.7%

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

       1. associate-*r* 7.7%

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

       2. +-commutative 7.7%

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

       3. *-commutative 7.7%

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

       4. *-commutative 7.7%

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

       5. associate-*l* 7.7%

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

       6. distribute-lft-out 7.7%

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

   11. Simplified 7.7%

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

   13. Applied egg-rr 31.4%

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

   if 1.9500000000000001e93 < im < 1.29999999999999994e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 8.5%

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

      1. +-commutative 8.5%

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

      2. unpow2 8.5%

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

      3. fma-define 8.5%

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

   5. Simplified 8.5%

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

   6. Taylor expanded in im around inf 8.5%

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

      1. *-commutative 8.5%

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

      2. associate-*r* 8.5%

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

   8. Simplified 8.5%

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

   9. Taylor expanded in re around 0 37.3%

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

       1. associate-*r* 37.3%

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

       2. +-commutative 37.3%

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

       3. *-commutative 37.3%

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

       4. *-commutative 37.3%

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

       5. associate-*l* 37.3%

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

       6. distribute-lft-out 37.3%

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

   11. Simplified 37.3%

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

   13. Applied egg-rr 41.7%

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

   if 1.29999999999999994e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 93.5%

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

   4. Taylor expanded in im around 0 93.5%

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

      1. +-commutative 93.5%

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

      2. unpow2 93.5%

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

      3. fma-define 93.5%

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

   6. Simplified 93.5%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00019:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00019)
   (cos re)
   (if (<= im 1.35e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00019) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = (0.5 * cos(re)) * pow(im, 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 <= 0.00019d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (0.5d0 * cos(re)) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00019) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00019:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00019)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00019)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (0.5 * cos(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00019], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.9000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.3%

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

   if 1.9000000000000001e-4 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 78.1%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00019:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.155:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.155)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.35e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.155) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.155)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.155], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.154999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 80.9%

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

      1. +-commutative 80.9%

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

      2. unpow2 80.9%

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

      3. fma-define 80.9%

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

   5. Simplified 80.9%

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

   if 0.154999999999999999 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 79.5%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.155:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.4e-5) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.4e-5) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.4d-5) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.4e-5) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.4e-5:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.4e-5)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.4e-5)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.4e-5], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.39999999999999945e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.3%

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

   if 9.39999999999999945e-5 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 84.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+93}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+115}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+153}:\\ \;\;\;\;-2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 260.0)
   (cos re)
   (if (<= im 1.2e+93)
     (pow re -2.0)
     (if (<= im 2.6e+106)
       (fma re re -2.0)
       (if (<= im 6.2e+115)
         (pow re -2.0)
         (if (<= im 2.7e+153) (- -2.0 (* re re)) (* 0.5 (fma im im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = cos(re);
	} else if (im <= 1.2e+93) {
		tmp = pow(re, -2.0);
	} else if (im <= 2.6e+106) {
		tmp = fma(re, re, -2.0);
	} else if (im <= 6.2e+115) {
		tmp = pow(re, -2.0);
	} else if (im <= 2.7e+153) {
		tmp = -2.0 - (re * re);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 260.0)
		tmp = cos(re);
	elseif (im <= 1.2e+93)
		tmp = re ^ -2.0;
	elseif (im <= 2.6e+106)
		tmp = fma(re, re, -2.0);
	elseif (im <= 6.2e+115)
		tmp = re ^ -2.0;
	elseif (im <= 2.7e+153)
		tmp = Float64(-2.0 - Float64(re * re));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 260.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.2e+93], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 2.6e+106], N[(re * re + -2.0), $MachinePrecision], If[LessEqual[im, 6.2e+115], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 2.7e+153], N[(-2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 260

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   if 260 < im < 1.20000000000000005e93 or 2.6000000000000002e106 < im < 6.2000000000000001e115

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. unpow2 4.2%

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

      3. fma-define 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in im around inf 4.2%

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

      1. *-commutative 4.2%

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

      2. associate-*r* 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in re around 0 7.7%

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

       1. associate-*r* 7.7%

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

       2. +-commutative 7.7%

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

       3. *-commutative 7.7%

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

       4. *-commutative 7.7%

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

       5. associate-*l* 7.7%

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

       6. distribute-lft-out 7.7%

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

   11. Simplified 7.7%

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

   13. Applied egg-rr 31.4%

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

   if 1.20000000000000005e93 < im < 2.6000000000000002e106

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.5%

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

      1. +-commutative 5.5%

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

      2. unpow2 5.5%

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

      3. fma-define 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in im around inf 5.5%

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

      1. *-commutative 5.5%

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

      2. associate-*r* 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in re around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. +-commutative 0.0%

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

       3. *-commutative 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*l* 0.0%

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

       6. distribute-lft-out 0.0%

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

   11. Simplified 0.0%

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

   13. Applied egg-rr 100.0%

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

   if 6.2000000000000001e115 < im < 2.7000000000000001e153

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 9.2%

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

      1. +-commutative 9.2%

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

      2. unpow2 9.2%

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

      3. fma-define 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in im around inf 9.2%

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

      1. *-commutative 9.2%

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

      2. associate-*r* 9.2%

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

   8. Simplified 9.2%

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

   9. Taylor expanded in re around 0 45.3%

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

       1. associate-*r* 45.3%

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

       2. +-commutative 45.3%

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

       3. *-commutative 45.3%

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

       4. *-commutative 45.3%

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

       5. associate-*l* 45.3%

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

       6. distribute-lft-out 45.3%

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

   11. Simplified 45.3%

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

   13. Applied egg-rr 23.6%

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

   if 2.7000000000000001e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 93.5%

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

   4. Taylor expanded in im around 0 93.5%

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

      1. +-commutative 93.5%

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

      2. unpow2 93.5%

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

      3. fma-define 93.5%

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

   6. Simplified 93.5%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+93}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+115}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+153}:\\ \;\;\;\;-2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 242:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+92}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+116}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+152}:\\ \;\;\;\;-2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 242.0)
   (cos re)
   (if (<= im 9.8e+92)
     (pow re -2.0)
     (if (<= im 4.2e+108)
       (fma re re -2.0)
       (if (<= im 1.4e+116)
         (pow re -2.0)
         (if (<= im 7.2e+152) (- -2.0 (* re re)) (* 0.5 (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 242.0) {
		tmp = cos(re);
	} else if (im <= 9.8e+92) {
		tmp = pow(re, -2.0);
	} else if (im <= 4.2e+108) {
		tmp = fma(re, re, -2.0);
	} else if (im <= 1.4e+116) {
		tmp = pow(re, -2.0);
	} else if (im <= 7.2e+152) {
		tmp = -2.0 - (re * re);
	} else {
		tmp = 0.5 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 242.0)
		tmp = cos(re);
	elseif (im <= 9.8e+92)
		tmp = re ^ -2.0;
	elseif (im <= 4.2e+108)
		tmp = fma(re, re, -2.0);
	elseif (im <= 1.4e+116)
		tmp = re ^ -2.0;
	elseif (im <= 7.2e+152)
		tmp = Float64(-2.0 - Float64(re * re));
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 242.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 9.8e+92], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 4.2e+108], N[(re * re + -2.0), $MachinePrecision], If[LessEqual[im, 1.4e+116], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 7.2e+152], N[(-2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 242

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   if 242 < im < 9.8000000000000003e92 or 4.20000000000000019e108 < im < 1.40000000000000002e116

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. unpow2 4.2%

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

      3. fma-define 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in im around inf 4.2%

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

      1. *-commutative 4.2%

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

      2. associate-*r* 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in re around 0 7.7%

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

       1. associate-*r* 7.7%

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

       2. +-commutative 7.7%

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

       3. *-commutative 7.7%

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

       4. *-commutative 7.7%

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

       5. associate-*l* 7.7%

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

       6. distribute-lft-out 7.7%

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

   11. Simplified 7.7%

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

   13. Applied egg-rr 31.4%

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

   if 9.8000000000000003e92 < im < 4.20000000000000019e108

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.5%

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

      1. +-commutative 5.5%

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

      2. unpow2 5.5%

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

      3. fma-define 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in im around inf 5.5%

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

      1. *-commutative 5.5%

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

      2. associate-*r* 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in re around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. +-commutative 0.0%

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

       3. *-commutative 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*l* 0.0%

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

       6. distribute-lft-out 0.0%

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

   11. Simplified 0.0%

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

   13. Applied egg-rr 100.0%

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

   if 1.40000000000000002e116 < im < 7.1999999999999998e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 9.2%

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

      1. +-commutative 9.2%

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

      2. unpow2 9.2%

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

      3. fma-define 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in im around inf 9.2%

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

      1. *-commutative 9.2%

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

      2. associate-*r* 9.2%

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

   8. Simplified 9.2%

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

   9. Taylor expanded in re around 0 45.3%

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

       1. associate-*r* 45.3%

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

       2. +-commutative 45.3%

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

       3. *-commutative 45.3%

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

       4. *-commutative 45.3%

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

       5. associate-*l* 45.3%

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

       6. distribute-lft-out 45.3%

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

   11. Simplified 45.3%

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

   13. Applied egg-rr 23.6%

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

   if 7.1999999999999998e152 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0 93.5%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 93.5%

          \[\leadsto \color{blue}{{im}^{2} \cdot 0.5} \]

   11. Simplified 93.5%

       \[\leadsto \color{blue}{{im}^{2} \cdot 0.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 242:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+92}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+116}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+152}:\\ \;\;\;\;-2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.56 \cdot 10^{+93}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+152}:\\ \;\;\;\;-2 - re \cdot re\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 260.0)
   (cos re)
   (if (<= im 1.56e+93)
     (pow re -2.0)
     (if (<= im 2.5e+106)
       (fma re re -2.0)
       (if (<= im 1.05e+152)
         (- -2.0 (* re re))
         (if (<= im 9.8e+179) (fma re re -2.0) (pow re -2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = cos(re);
	} else if (im <= 1.56e+93) {
		tmp = pow(re, -2.0);
	} else if (im <= 2.5e+106) {
		tmp = fma(re, re, -2.0);
	} else if (im <= 1.05e+152) {
		tmp = -2.0 - (re * re);
	} else if (im <= 9.8e+179) {
		tmp = fma(re, re, -2.0);
	} else {
		tmp = pow(re, -2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 260.0)
		tmp = cos(re);
	elseif (im <= 1.56e+93)
		tmp = re ^ -2.0;
	elseif (im <= 2.5e+106)
		tmp = fma(re, re, -2.0);
	elseif (im <= 1.05e+152)
		tmp = Float64(-2.0 - Float64(re * re));
	elseif (im <= 9.8e+179)
		tmp = fma(re, re, -2.0);
	else
		tmp = re ^ -2.0;
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 260.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.56e+93], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 2.5e+106], N[(re * re + -2.0), $MachinePrecision], If[LessEqual[im, 1.05e+152], N[(-2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.8e+179], N[(re * re + -2.0), $MachinePrecision], N[Power[re, -2.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 260

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   if 260 < im < 1.56000000000000004e93 or 9.7999999999999997e179 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 57.2%

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

      1. +-commutative 57.2%

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

      2. unpow2 57.2%

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

      3. fma-define 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in im around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r* 57.2%

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

   8. Simplified 57.2%

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

   9. Taylor expanded in re around 0 3.5%

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

       1. associate-*r* 3.5%

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

       2. +-commutative 3.5%

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

       3. *-commutative 3.5%

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

       4. *-commutative 3.5%

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

       5. associate-*l* 3.5%

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

       6. distribute-lft-out 24.7%

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

   11. Simplified 24.7%

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

   13. Applied egg-rr 25.6%

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

   if 1.56000000000000004e93 < im < 2.4999999999999999e106 or 1.0500000000000001e152 < im < 9.7999999999999997e179

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. unpow2 58.8%

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

      3. fma-define 58.8%

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

   5. Simplified 58.8%

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

   6. Taylor expanded in im around inf 58.8%

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

      1. *-commutative 58.8%

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

      2. associate-*r* 58.8%

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

   8. Simplified 58.8%

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

   9. Taylor expanded in re around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. +-commutative 0.0%

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

       3. *-commutative 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*l* 0.0%

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

       6. distribute-lft-out 33.3%

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

   11. Simplified 33.3%

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

   13. Applied egg-rr 56.4%

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

   if 2.4999999999999999e106 < im < 1.0500000000000001e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 8.9%

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

      1. +-commutative 8.9%

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

      2. unpow2 8.9%

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

      3. fma-define 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in im around inf 8.9%

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

      1. *-commutative 8.9%

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

      2. associate-*r* 8.9%

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

   8. Simplified 8.9%

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

   9. Taylor expanded in re around 0 48.8%

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

       1. associate-*r* 48.8%

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

       2. +-commutative 48.8%

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

       3. *-commutative 48.8%

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

       4. *-commutative 48.8%

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

       5. associate-*l* 48.8%

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

       6. distribute-lft-out 48.8%

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

   11. Simplified 48.8%

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

   13. Applied egg-rr 25.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.56 \cdot 10^{+93}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+152}:\\ \;\;\;\;-2 - re \cdot re\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -2\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{+92}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+153}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (cos re)
   (if (<= im 8.4e+92)
     (pow re -2.0)
     (if (<= im 1.4e+153)
       (+ 0.25 (* 0.25 (pow re 2.0)))
       (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re);
	} else if (im <= 8.4e+92) {
		tmp = pow(re, -2.0);
	} else if (im <= 1.4e+153) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 8.4e+92)
		tmp = re ^ -2.0;
	elseif (im <= 1.4e+153)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8.4e+92], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 1.4e+153], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 250

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   if 250 < im < 8.39999999999999944e92

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. unpow2 4.2%

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

      3. fma-define 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in im around inf 4.2%

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

      1. *-commutative 4.2%

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

      2. associate-*r* 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in re around 0 7.7%

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

       1. associate-*r* 7.7%

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

       2. +-commutative 7.7%

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

       3. *-commutative 7.7%

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

       4. *-commutative 7.7%

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

       5. associate-*l* 7.7%

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

       6. distribute-lft-out 7.7%

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

   11. Simplified 7.7%

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

   13. Applied egg-rr 31.4%

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

   if 8.39999999999999944e92 < im < 1.39999999999999993e153

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.2%

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

   5. Taylor expanded in re around 0 30.8%

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

      1. *-commutative 30.8%

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

   7. Simplified 30.8%

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

   if 1.39999999999999993e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 93.5%

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

   4. Taylor expanded in im around 0 93.5%

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

      1. +-commutative 93.5%

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

      2. unpow2 93.5%

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

      3. fma-define 93.5%

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

   6. Simplified 93.5%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{+92}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+153}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+116} \lor \neg \left(im \leq 2.85 \cdot 10^{+165}\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 260.0)
   (cos re)
   (if (or (<= im 1.1e+116) (not (<= im 2.85e+165)))
     (pow re -2.0)
     (- -2.0 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = cos(re);
	} else if ((im <= 1.1e+116) || !(im <= 2.85e+165)) {
		tmp = pow(re, -2.0);
	} else {
		tmp = -2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 260.0d0) then
        tmp = cos(re)
    else if ((im <= 1.1d+116) .or. (.not. (im <= 2.85d+165))) then
        tmp = re ** (-2.0d0)
    else
        tmp = (-2.0d0) - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = Math.cos(re);
	} else if ((im <= 1.1e+116) || !(im <= 2.85e+165)) {
		tmp = Math.pow(re, -2.0);
	} else {
		tmp = -2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 260.0:
		tmp = math.cos(re)
	elif (im <= 1.1e+116) or not (im <= 2.85e+165):
		tmp = math.pow(re, -2.0)
	else:
		tmp = -2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 260.0)
		tmp = cos(re);
	elseif ((im <= 1.1e+116) || !(im <= 2.85e+165))
		tmp = re ^ -2.0;
	else
		tmp = Float64(-2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 260.0)
		tmp = cos(re);
	elseif ((im <= 1.1e+116) || ~((im <= 2.85e+165)))
		tmp = re ^ -2.0;
	else
		tmp = -2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 260.0], N[Cos[re], $MachinePrecision], If[Or[LessEqual[im, 1.1e+116], N[Not[LessEqual[im, 2.85e+165]], $MachinePrecision]], N[Power[re, -2.0], $MachinePrecision], N[(-2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 260

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   if 260 < im < 1.1e116 or 2.85000000000000013e165 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 56.7%

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

      1. +-commutative 56.7%

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

      2. unpow2 56.7%

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

      3. fma-define 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in im around inf 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-*r* 56.7%

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

   8. Simplified 56.7%

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

   9. Taylor expanded in re around 0 3.1%

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

       1. associate-*r* 3.1%

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

       2. +-commutative 3.1%

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

       3. *-commutative 3.1%

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

       4. *-commutative 3.1%

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

       5. associate-*l* 3.1%

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

       6. distribute-lft-out 23.8%

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

   11. Simplified 23.8%

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

   13. Applied egg-rr 23.0%

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

   if 1.1e116 < im < 2.85000000000000013e165

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 20.6%

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

      1. +-commutative 20.6%

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

      2. unpow2 20.6%

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

      3. fma-define 20.6%

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

   5. Simplified 20.6%

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

   6. Taylor expanded in im around inf 20.6%

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

      1. *-commutative 20.6%

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

      2. associate-*r* 20.6%

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

   8. Simplified 20.6%

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

   9. Taylor expanded in re around 0 39.7%

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

       1. associate-*r* 39.7%

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

       2. +-commutative 39.7%

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

       3. *-commutative 39.7%

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

       4. *-commutative 39.7%

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

       5. associate-*l* 39.7%

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

       6. distribute-lft-out 52.2%

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

   11. Simplified 52.2%

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

   13. Applied egg-rr 20.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+116} \lor \neg \left(im \leq 2.85 \cdot 10^{+165}\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-2 - re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;-2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8e+18) (cos re) (- -2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+18) {
		tmp = cos(re);
	} else {
		tmp = -2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.8d+18) then
        tmp = cos(re)
    else
        tmp = (-2.0d0) - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+18) {
		tmp = Math.cos(re);
	} else {
		tmp = -2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.8e+18:
		tmp = math.cos(re)
	else:
		tmp = -2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8e+18)
		tmp = cos(re);
	else
		tmp = Float64(-2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.8e+18)
		tmp = cos(re);
	else
		tmp = -2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8e+18], N[Cos[re], $MachinePrecision], N[(-2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.8e18

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 57.2%

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

   if 3.8e18 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 51.8%

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

      1. +-commutative 51.8%

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

      2. unpow2 51.8%

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

      3. fma-define 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in im around inf 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*r* 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in re around 0 12.3%

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

       1. associate-*r* 12.3%

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

       2. +-commutative 12.3%

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

       3. *-commutative 12.3%

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

       4. *-commutative 12.3%

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

       5. associate-*l* 12.3%

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

       6. distribute-lft-out 32.6%

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

   11. Simplified 32.6%

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

   13. Applied egg-rr 8.7%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;-2 - re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.1% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.8e+20) 1.0 (- -2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.8e+20) {
		tmp = 1.0;
	} else {
		tmp = -2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.8d+20) then
        tmp = 1.0d0
    else
        tmp = (-2.0d0) - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.8e+20) {
		tmp = 1.0;
	} else {
		tmp = -2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.8e+20:
		tmp = 1.0
	else:
		tmp = -2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.8e+20)
		tmp = 1.0;
	else
		tmp = Float64(-2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.8e+20)
		tmp = 1.0;
	else
		tmp = -2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.8e+20], 1.0, N[(-2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.8e20

   1. Initial program 100.0%

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

   4. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{\cos re \cdot -2}{\cos re \cdot -2 + \left(\cos re \cdot -2 - \cos re \cdot -2\right)}} \]
   5. Step-by-step derivation

      1. +-inverses 33.8%

         \[\leadsto \frac{\cos re \cdot -2}{\cos re \cdot -2 + \color{blue}{0}} \]

      2. +-rgt-identity 33.8%

         \[\leadsto \frac{\cos re \cdot -2}{\color{blue}{\cos re \cdot -2}} \]

      3. *-inverses 33.8%

         \[\leadsto \color{blue}{1} \]

   6. Simplified 33.8%

      \[\leadsto \color{blue}{1} \]

   if 4.8e20 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 51.8%

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

      1. +-commutative 51.8%

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

      2. unpow2 51.8%

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

      3. fma-define 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in im around inf 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*r* 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in re around 0 12.3%

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

       1. associate-*r* 12.3%

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

       2. +-commutative 12.3%

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

       3. *-commutative 12.3%

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

       4. *-commutative 12.3%

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

       5. associate-*l* 12.3%

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

       6. distribute-lft-out 32.6%

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

   11. Simplified 32.6%

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

   13. Applied egg-rr 8.7%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-2 - re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 2.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
5. Step-by-step derivation

   1. pow-base-1 2.2%

      \[\leadsto \log \color{blue}{1} \]

   2. metadata-eval 2.2%

      \[\leadsto \color{blue}{0} \]

6. Simplified 2.2%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 2.2%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 14: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 7.7%

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

5. Taylor expanded in re around 0 7.6%

   \[\leadsto \color{blue}{0.25} \]

6. Final simplification 7.6%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 15: 28.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 26.0%

   \[\leadsto \color{blue}{\frac{\cos re \cdot -2}{\cos re \cdot -2 + \left(\cos re \cdot -2 - \cos re \cdot -2\right)}} \]
5. Step-by-step derivation

   1. +-inverses 26.0%

      \[\leadsto \frac{\cos re \cdot -2}{\cos re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 26.0%

      \[\leadsto \frac{\cos re \cdot -2}{\color{blue}{\cos re \cdot -2}} \]

   3. *-inverses 26.0%

      \[\leadsto \color{blue}{1} \]

6. Simplified 26.0%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 26.0%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))