math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 11

Speedup: 1.0×

• 49.7% 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: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 640 \lor \neg \left(im \leq 1.1 \cdot 10^{+141}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 640.0) (not (<= im 1.1e+141)))
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (+ 1.0 (log1p (expm1 (* (pow re 2.0) -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 640.0) || !(im <= 1.1e+141)) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else {
		tmp = 1.0 + log1p(expm1((pow(re, 2.0) * -0.5)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 640.0) || !(im <= 1.1e+141))
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	else
		tmp = Float64(1.0 + log1p(expm1(Float64((re ^ 2.0) * -0.5))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 640.0], N[Not[LessEqual[im, 1.1e+141]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[1 + N[(Exp[N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 640 or 1.1e141 < 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 88.0%

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

      1. +-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. fma-define 88.0%

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

   5. Simplified 88.0%

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

   if 640 < im < 1.1e141

   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 5.0%

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

      1. +-commutative 5.0%

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

      2. unpow2 5.0%

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

      3. fma-define 5.0%

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

   5. Simplified 5.0%

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

   6. Taylor expanded in re around 0 17.6%

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

      1. associate-*r* 17.6%

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

      2. distribute-rgt-out 17.6%

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

      3. +-commutative 17.6%

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

      4. unpow2 17.6%

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

      5. fma-undefine 17.6%

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

      6. +-commutative 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in im around 0 8.3%

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

       1. distribute-rgt-in 8.3%

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

       2. metadata-eval 8.3%

          \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

       3. *-commutative 8.3%

          \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

       4. associate-*l* 8.3%

          \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

       5. metadata-eval 8.3%

          \[\leadsto 1 + {re}^{2} \cdot \color{blue}{-0.5} \]

   11. Simplified 8.3%

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

       1. log1p-expm1-u 25.1%

          \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]

   13. Applied egg-rr 25.1%

       \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 640 \lor \neg \left(im \leq 1.1 \cdot 10^{+141}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{log1p}\left(\mathsf{expm1}\left({re}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + {re}^{2} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 2.3e-6)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.35e+154)
       (* (fma im im 2.0) (+ 0.5 (* (pow re 2.0) -0.25)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 2.3e-6) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = fma(im, im, 2.0) * (0.5 + (pow(re, 2.0) * -0.25));
	} 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 <= 2.3e-6)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 + Float64((re ^ 2.0) * -0.25)));
	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, 2.3e-6], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e-6

   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 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. fma-define 87.2%

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

   5. Simplified 87.2%

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

   if 2.3e-6 < 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 im around 0 9.6%

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

      1. +-commutative 9.6%

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

      2. unpow2 9.6%

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

      3. fma-define 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in re around 0 20.3%

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

      1. associate-*r* 20.3%

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

      2. distribute-rgt-out 20.3%

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

      3. +-commutative 20.3%

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

      4. unpow2 20.3%

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

      5. fma-undefine 20.3%

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

      6. +-commutative 20.3%

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

   8. Simplified 20.3%

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

   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%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + {re}^{2} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6600000000 \lor \neg \left(im \leq 8.5 \cdot 10^{+118}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{{re}^{4} \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 6600000000.0) (not (<= im 8.5e+118)))
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (+ 1.0 (sqrt (* (pow re 4.0) 0.25)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 6600000000.0) || !(im <= 8.5e+118)) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else {
		tmp = 1.0 + sqrt((pow(re, 4.0) * 0.25));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 6600000000.0) || !(im <= 8.5e+118))
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	else
		tmp = Float64(1.0 + sqrt(Float64((re ^ 4.0) * 0.25)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 6600000000.0], N[Not[LessEqual[im, 8.5e+118]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Sqrt[N[(N[Power[re, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.6e9 or 8.50000000000000033e118 < 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 84.8%

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

      1. +-commutative 84.8%

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

      2. unpow2 84.8%

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

      3. fma-define 84.8%

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

   5. Simplified 84.8%

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

   if 6.6e9 < im < 8.50000000000000033e118

   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 4.7%

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

      1. +-commutative 4.7%

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

      2. unpow2 4.7%

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

      3. fma-define 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in re around 0 10.7%

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

      1. associate-*r* 10.7%

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

      2. distribute-rgt-out 10.7%

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

      3. +-commutative 10.7%

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

      4. unpow2 10.7%

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

      5. fma-undefine 10.7%

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

      6. +-commutative 10.7%

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

   8. Simplified 10.7%

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

   9. Taylor expanded in im around 0 6.2%

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

       1. distribute-rgt-in 6.2%

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

       2. metadata-eval 6.2%

          \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

       3. *-commutative 6.2%

          \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

       4. associate-*l* 6.2%

          \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

       5. metadata-eval 6.2%

          \[\leadsto 1 + {re}^{2} \cdot \color{blue}{-0.5} \]

   11. Simplified 6.2%

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

       1. add-sqr-sqrt 0.8%

          \[\leadsto 1 + \color{blue}{\sqrt{{re}^{2} \cdot -0.5} \cdot \sqrt{{re}^{2} \cdot -0.5}} \]

       2. sqrt-unprod 29.8%

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

       3. swap-sqr 29.8%

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

       4. pow-prod-up 29.8%

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

       5. metadata-eval 29.8%

          \[\leadsto 1 + \sqrt{{re}^{\color{blue}{4}} \cdot \left(-0.5 \cdot -0.5\right)} \]

       6. metadata-eval 29.8%

          \[\leadsto 1 + \sqrt{{re}^{4} \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 29.8%

       \[\leadsto 1 + \color{blue}{\sqrt{{re}^{4} \cdot 0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6600000000 \lor \neg \left(im \leq 8.5 \cdot 10^{+118}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{{re}^{4} \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.3e-6)
   (cos re)
   (if (<= im 3.8e+133)
     (+ 1.0 (* (pow re 2.0) -0.5))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.3e-6) {
		tmp = cos(re);
	} else if (im <= 3.8e+133) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} 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 <= 2.3d-6) then
        tmp = cos(re)
    else if (im <= 3.8d+133) then
        tmp = 1.0d0 + ((re ** 2.0d0) * (-0.5d0))
    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 <= 2.3e-6) {
		tmp = Math.cos(re);
	} else if (im <= 3.8e+133) {
		tmp = 1.0 + (Math.pow(re, 2.0) * -0.5);
	} else {
		tmp = (0.5 * Math.cos(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.3e-6:
		tmp = math.cos(re)
	elif im <= 3.8e+133:
		tmp = 1.0 + (math.pow(re, 2.0) * -0.5)
	else:
		tmp = (0.5 * math.cos(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.3e-6)
		tmp = cos(re);
	elseif (im <= 3.8e+133)
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	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 <= 2.3e-6)
		tmp = cos(re);
	elseif (im <= 3.8e+133)
		tmp = 1.0 + ((re ^ 2.0) * -0.5);
	else
		tmp = (0.5 * cos(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.3e-6], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.8e+133], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $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 < 2.3e-6

   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 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. fma-define 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in im around 0 73.1%

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

   if 2.3e-6 < im < 3.8000000000000002e133

   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 9.8%

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

      1. +-commutative 9.8%

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

      2. unpow2 9.8%

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

      3. fma-define 9.8%

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

   5. Simplified 9.8%

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

   6. Taylor expanded in re around 0 19.9%

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

      1. associate-*r* 19.9%

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

      2. distribute-rgt-out 19.9%

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

      3. +-commutative 19.9%

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

      4. unpow2 19.9%

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

      5. fma-undefine 19.9%

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

      6. +-commutative 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in im around 0 12.0%

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

       1. distribute-rgt-in 12.0%

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

       2. metadata-eval 12.0%

          \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

       3. *-commutative 12.0%

          \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

       4. associate-*l* 12.0%

          \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

       5. metadata-eval 12.0%

          \[\leadsto 1 + {re}^{2} \cdot \color{blue}{-0.5} \]

   11. Simplified 12.0%

       \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]

   if 3.8000000000000002e133 < 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 85.5%

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

      1. +-commutative 85.5%

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

      2. unpow2 85.5%

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

      3. fma-define 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in im around inf 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-*l* 85.5%

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

      3. *-commutative 85.5%

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

   8. Simplified 85.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1950000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+118}:\\ \;\;\;\;1 + \sqrt{{re}^{4} \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1950000.0)
   (cos re)
   (if (<= im 5.2e+118)
     (+ 1.0 (sqrt (* (pow re 4.0) 0.25)))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1950000.0) {
		tmp = cos(re);
	} else if (im <= 5.2e+118) {
		tmp = 1.0 + sqrt((pow(re, 4.0) * 0.25));
	} 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 <= 1950000.0d0) then
        tmp = cos(re)
    else if (im <= 5.2d+118) then
        tmp = 1.0d0 + sqrt(((re ** 4.0d0) * 0.25d0))
    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 <= 1950000.0) {
		tmp = Math.cos(re);
	} else if (im <= 5.2e+118) {
		tmp = 1.0 + Math.sqrt((Math.pow(re, 4.0) * 0.25));
	} else {
		tmp = (0.5 * Math.cos(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1950000.0:
		tmp = math.cos(re)
	elif im <= 5.2e+118:
		tmp = 1.0 + math.sqrt((math.pow(re, 4.0) * 0.25))
	else:
		tmp = (0.5 * math.cos(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1950000.0)
		tmp = cos(re);
	elseif (im <= 5.2e+118)
		tmp = Float64(1.0 + sqrt(Float64((re ^ 4.0) * 0.25)));
	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 <= 1950000.0)
		tmp = cos(re);
	elseif (im <= 5.2e+118)
		tmp = 1.0 + sqrt(((re ^ 4.0) * 0.25));
	else
		tmp = (0.5 * cos(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1950000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.2e+118], N[(1.0 + N[Sqrt[N[(N[Power[re, 4.0], $MachinePrecision] * 0.25), $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 < 1.95e6

   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 86.8%

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

      1. +-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. fma-define 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in im around 0 72.6%

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

   if 1.95e6 < im < 5.20000000000000032e118

   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 4.6%

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

      1. +-commutative 4.6%

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

      2. unpow2 4.6%

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

      3. fma-define 4.6%

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

   5. Simplified 4.6%

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

   6. Taylor expanded in re around 0 10.2%

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

      1. associate-*r* 10.2%

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

      2. distribute-rgt-out 10.2%

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

      3. +-commutative 10.2%

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

      4. unpow2 10.2%

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

      5. fma-undefine 10.2%

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

      6. +-commutative 10.2%

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

   8. Simplified 10.2%

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

   9. Taylor expanded in im around 0 6.0%

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

       1. distribute-rgt-in 6.0%

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

       2. metadata-eval 6.0%

          \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

       3. *-commutative 6.0%

          \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

       4. associate-*l* 6.0%

          \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

       5. metadata-eval 6.0%

          \[\leadsto 1 + {re}^{2} \cdot \color{blue}{-0.5} \]

   11. Simplified 6.0%

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

       1. add-sqr-sqrt 0.8%

          \[\leadsto 1 + \color{blue}{\sqrt{{re}^{2} \cdot -0.5} \cdot \sqrt{{re}^{2} \cdot -0.5}} \]

       2. sqrt-unprod 27.8%

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

       3. swap-sqr 27.8%

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

       4. pow-prod-up 27.8%

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

       5. metadata-eval 27.8%

          \[\leadsto 1 + \sqrt{{re}^{\color{blue}{4}} \cdot \left(-0.5 \cdot -0.5\right)} \]

       6. metadata-eval 27.8%

          \[\leadsto 1 + \sqrt{{re}^{4} \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 27.8%

       \[\leadsto 1 + \color{blue}{\sqrt{{re}^{4} \cdot 0.25}} \]

   if 5.20000000000000032e118 < 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 80.5%

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

      1. +-commutative 80.5%

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

      2. unpow2 80.5%

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

      3. fma-define 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in im around inf 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l* 80.5%

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

      3. *-commutative 80.5%

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

   8. Simplified 80.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1950000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+118}:\\ \;\;\;\;1 + \sqrt{{re}^{4} \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+140}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.3e-6)
   (cos re)
   (if (<= im 5.2e+140)
     (+ 1.0 (* (pow re 2.0) -0.5))
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.3e-6) {
		tmp = cos(re);
	} else if (im <= 5.2e+140) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.3e-6)
		tmp = cos(re);
	elseif (im <= 5.2e+140)
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.3e-6], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.2e+140], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e-6

   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 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. fma-define 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in im around 0 73.1%

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

   if 2.3e-6 < im < 5.2000000000000002e140

   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 9.6%

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

      1. +-commutative 9.6%

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

      2. unpow2 9.6%

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

      3. fma-define 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in re around 0 21.5%

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

      1. associate-*r* 21.5%

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

      2. distribute-rgt-out 21.5%

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

      3. +-commutative 21.5%

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

      4. unpow2 21.5%

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

      5. fma-undefine 21.5%

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

      6. +-commutative 21.5%

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

   8. Simplified 21.5%

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

   9. Taylor expanded in im around 0 11.3%

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

       1. distribute-rgt-in 11.3%

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

       2. metadata-eval 11.3%

          \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

       3. *-commutative 11.3%

          \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

       4. associate-*l* 11.3%

          \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

       5. metadata-eval 11.3%

          \[\leadsto 1 + {re}^{2} \cdot \color{blue}{-0.5} \]

   11. Simplified 11.3%

       \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]

   if 5.2000000000000002e140 < 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 91.2%

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

      1. +-commutative 91.2%

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

      2. unpow2 91.2%

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

      3. fma-define 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in re around 0 76.3%

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

      1. +-commutative 76.3%

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

      2. unpow2 76.3%

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

      3. fma-undefine 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+140}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 780:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 780.0)
   (cos re)
   (if (<= im 2.35e+138) (* (pow re 2.0) -0.5) (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 780.0) {
		tmp = cos(re);
	} else if (im <= 2.35e+138) {
		tmp = pow(re, 2.0) * -0.5;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 780.0)
		tmp = cos(re);
	elseif (im <= 2.35e+138)
		tmp = Float64((re ^ 2.0) * -0.5);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 780.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.35e+138], N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 780

   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 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. fma-define 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in im around 0 73.0%

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

   if 780 < im < 2.3499999999999999e138

   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 5.0%

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

      1. +-commutative 5.0%

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

      2. unpow2 5.0%

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

      3. fma-define 5.0%

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

   5. Simplified 5.0%

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

   6. Taylor expanded in re around 0 17.6%

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

      1. associate-*r* 17.6%

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

      2. distribute-rgt-out 17.6%

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

      3. +-commutative 17.6%

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

      4. unpow2 17.6%

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

      5. fma-undefine 17.6%

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

      6. +-commutative 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in re around inf 15.9%

      \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2}\right)} \]

   10. Taylor expanded in im around 0 7.5%

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

   if 2.3499999999999999e138 < 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 91.2%

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

      1. +-commutative 91.2%

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

      2. unpow2 91.2%

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

      3. fma-define 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in re around 0 76.3%

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

      1. +-commutative 76.3%

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

      2. unpow2 76.3%

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

      3. fma-undefine 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 780:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 620.0)
   (cos re)
   (if (<= im 1.1e+141) (* (pow re 2.0) -0.5) (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 620.0) {
		tmp = cos(re);
	} else if (im <= 1.1e+141) {
		tmp = pow(re, 2.0) * -0.5;
	} else {
		tmp = 0.5 * 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 <= 620.0d0) then
        tmp = cos(re)
    else if (im <= 1.1d+141) then
        tmp = (re ** 2.0d0) * (-0.5d0)
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 620.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.1e+141) {
		tmp = Math.pow(re, 2.0) * -0.5;
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 620.0:
		tmp = math.cos(re)
	elif im <= 1.1e+141:
		tmp = math.pow(re, 2.0) * -0.5
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 620.0)
		tmp = cos(re);
	elseif (im <= 1.1e+141)
		tmp = Float64((re ^ 2.0) * -0.5);
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 620.0)
		tmp = cos(re);
	elseif (im <= 1.1e+141)
		tmp = (re ^ 2.0) * -0.5;
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 620.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.1e+141], N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 620

   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 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. fma-define 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in im around 0 73.0%

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

   if 620 < im < 1.1e141

   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 5.0%

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

      1. +-commutative 5.0%

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

      2. unpow2 5.0%

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

      3. fma-define 5.0%

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

   5. Simplified 5.0%

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

   6. Taylor expanded in re around 0 17.6%

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

      1. associate-*r* 17.6%

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

      2. distribute-rgt-out 17.6%

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

      3. +-commutative 17.6%

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

      4. unpow2 17.6%

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

      5. fma-undefine 17.6%

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

      6. +-commutative 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in re around inf 15.9%

      \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2}\right)} \]

   10. Taylor expanded in im around 0 7.5%

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

   if 1.1e141 < 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 91.2%

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

      1. +-commutative 91.2%

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

      2. unpow2 91.2%

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

      3. fma-define 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in im around inf 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-*l* 91.2%

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

      3. *-commutative 91.2%

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

   8. Simplified 91.2%

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

   9. Taylor expanded in re around 0 76.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700.0) (cos re) (* (pow re 2.0) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = cos(re);
	} else {
		tmp = pow(re, 2.0) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 700.0d0) then
        tmp = cos(re)
    else
        tmp = (re ** 2.0d0) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.pow(re, 2.0) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700.0:
		tmp = math.cos(re)
	else:
		tmp = math.pow(re, 2.0) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		tmp = cos(re);
	else
		tmp = Float64((re ^ 2.0) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 700.0)
		tmp = cos(re);
	else
		tmp = (re ^ 2.0) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], N[Cos[re], $MachinePrecision], N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 700

   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 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. fma-define 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in im around 0 73.0%

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

   if 700 < 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 52.1%

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

      1. +-commutative 52.1%

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

      2. unpow2 52.1%

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

      3. fma-define 52.1%

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

   5. Simplified 52.1%

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

   6. Taylor expanded in re around 0 8.5%

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

      1. associate-*r* 8.5%

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

      2. distribute-rgt-out 37.8%

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

      3. +-commutative 37.8%

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

      4. unpow2 37.8%

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

      5. fma-undefine 37.8%

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

      6. +-commutative 37.8%

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

   8. Simplified 37.8%

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

   9. Taylor expanded in re around inf 15.4%

      \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2}\right)} \]

   10. Taylor expanded in im around 0 5.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.9% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (cos re))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.cos(re)

Julia

function code(re, im)
	return cos(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[re], $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. Taylor expanded in im around 0 77.0%

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

   1. +-commutative 77.0%

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

   2. unpow2 77.0%

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

   3. fma-define 77.0%

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

5. Simplified 77.0%

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

6. Taylor expanded in im around 0 52.5%

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

7. Final simplification 52.5%

   \[\leadsto \cos re \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))