math.cos on complex, real part

Percentage Accurate: 100.0% → 99.7%

Time: 7.0s

Alternatives: 13

Speedup: 1.0×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im\_m} + e^{im\_m} \leq 4:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im\_m} + 3\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im_m)) (exp im_m)) 4.0)
     (* t_0 (fma im_m im_m 2.0))
     (* t_0 (+ (exp im_m) 3.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((exp(-im_m) + exp(im_m)) <= 4.0) {
		tmp = t_0 * fma(im_m, im_m, 2.0);
	} else {
		tmp = t_0 * (exp(im_m) + 3.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) + exp(im_m)) <= 4.0)
		tmp = Float64(t_0 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(t_0 * Float64(exp(im_m) + 3.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[(t$95$0 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 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 0 100.0%

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

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   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 51.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;e^{-im\_m} + e^{im\_m} \leq 4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im\_m} + 3\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ (exp (- im_m)) (exp im_m)) 4.0)
   (cos re)
   (* (* 0.5 (cos re)) (+ (exp im_m) 3.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((exp(-im_m) + exp(im_m)) <= 4.0) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * cos(re)) * (exp(im_m) + 3.0);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(-im_m) + exp(im_m)) <= 4.0d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * cos(re)) * (exp(im_m) + 3.0d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) + Math.exp(im_m)) <= 4.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp(im_m) + 3.0);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (math.exp(-im_m) + math.exp(im_m)) <= 4.0:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp(im_m) + 3.0)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) + exp(im_m)) <= 4.0)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(im_m) + 3.0));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) + exp(im_m)) <= 4.0)
		tmp = cos(re);
	else
		tmp = (0.5 * cos(re)) * (exp(im_m) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 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 99.8%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   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 51.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im_m)) (exp im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (0.5 * cos(re)) * (exp(-im_m) + exp(im_m));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $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. Add Preprocessing

Alternative 4: 94.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.26 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im\_m} + \left(1 + im\_m \cdot \left(im\_m \cdot \left(0.5 + im\_m \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.26e-6)
   (cos re)
   (if (<= im_m 1.02e+103)
     (*
      0.5
      (+
       (exp im_m)
       (+
        1.0
        (* im_m (+ (* im_m (+ 0.5 (* im_m -0.16666666666666666))) -1.0)))))
     (*
      (* 0.5 (cos re))
      (+
       4.0
       (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.26e-6) {
		tmp = cos(re);
	} else if (im_m <= 1.02e+103) {
		tmp = 0.5 * (exp(im_m) + (1.0 + (im_m * ((im_m * (0.5 + (im_m * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.26d-6) then
        tmp = cos(re)
    else if (im_m <= 1.02d+103) then
        tmp = 0.5d0 * (exp(im_m) + (1.0d0 + (im_m * ((im_m * (0.5d0 + (im_m * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.26e-6) {
		tmp = Math.cos(re);
	} else if (im_m <= 1.02e+103) {
		tmp = 0.5 * (Math.exp(im_m) + (1.0 + (im_m * ((im_m * (0.5 + (im_m * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.26e-6:
		tmp = math.cos(re)
	elif im_m <= 1.02e+103:
		tmp = 0.5 * (math.exp(im_m) + (1.0 + (im_m * ((im_m * (0.5 + (im_m * -0.16666666666666666))) + -1.0))))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.26e-6)
		tmp = cos(re);
	elseif (im_m <= 1.02e+103)
		tmp = Float64(0.5 * Float64(exp(im_m) + Float64(1.0 + Float64(im_m * Float64(Float64(im_m * Float64(0.5 + Float64(im_m * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.26e-6)
		tmp = cos(re);
	elseif (im_m <= 1.02e+103)
		tmp = 0.5 * (exp(im_m) + (1.0 + (im_m * ((im_m * (0.5 + (im_m * -0.16666666666666666))) + -1.0))));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.26e-6], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 1.02e+103], N[(0.5 * N[(N[Exp[im$95$m], $MachinePrecision] + N[(1.0 + N[(im$95$m * N[(N[(im$95$m * N[(0.5 + N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.26000000000000001e-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 65.2%

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

   if 1.26000000000000001e-6 < im < 1.01999999999999991e103

   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 83.4%

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

   4. Taylor expanded in im around 0 83.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + -0.16666666666666666 \cdot im\right) - 1\right)\right)} + e^{im}\right) \]

   if 1.01999999999999991e103 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.26 \cdot 10^{-6}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 2.4:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im\_m} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im\_m \cdot \left(im\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 2.4)
   (cos re)
   (if (<= im_m 2.7e+154)
     (+ (* 0.5 (exp im_m)) 1.5)
     (* (cos re) (+ 2.0 (* im_m (* im_m 0.25)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.4) {
		tmp = cos(re);
	} else if (im_m <= 2.7e+154) {
		tmp = (0.5 * exp(im_m)) + 1.5;
	} else {
		tmp = cos(re) * (2.0 + (im_m * (im_m * 0.25)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.4d0) then
        tmp = cos(re)
    else if (im_m <= 2.7d+154) then
        tmp = (0.5d0 * exp(im_m)) + 1.5d0
    else
        tmp = cos(re) * (2.0d0 + (im_m * (im_m * 0.25d0)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.4) {
		tmp = Math.cos(re);
	} else if (im_m <= 2.7e+154) {
		tmp = (0.5 * Math.exp(im_m)) + 1.5;
	} else {
		tmp = Math.cos(re) * (2.0 + (im_m * (im_m * 0.25)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 2.4:
		tmp = math.cos(re)
	elif im_m <= 2.7e+154:
		tmp = (0.5 * math.exp(im_m)) + 1.5
	else:
		tmp = math.cos(re) * (2.0 + (im_m * (im_m * 0.25)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 2.4)
		tmp = cos(re);
	elseif (im_m <= 2.7e+154)
		tmp = Float64(Float64(0.5 * exp(im_m)) + 1.5);
	else
		tmp = Float64(cos(re) * Float64(2.0 + Float64(im_m * Float64(im_m * 0.25))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 2.4)
		tmp = cos(re);
	elseif (im_m <= 2.7e+154)
		tmp = (0.5 * exp(im_m)) + 1.5;
	else
		tmp = cos(re) * (2.0 + (im_m * (im_m * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 2.4], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 2.7e+154], N[(N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im$95$m * N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.39999999999999991

   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 65.3%

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

   if 2.39999999999999991 < im < 2.70000000000000006e154

   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 100.0%

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

   5. Taylor expanded in re around 0 85.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 85.3%

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

      2. distribute-lft-in 85.3%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 85.3%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 85.3%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   if 2.70000000000000006e154 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \cos re \cdot \left(2 + im \cdot \color{blue}{\left(im \cdot 0.25\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 2.1:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im\_m} + 1.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 2.1) (cos re) (+ (* 0.5 (exp im_m)) 1.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.1) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * exp(im_m)) + 1.5;
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.1d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * exp(im_m)) + 1.5d0
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.1) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.exp(im_m)) + 1.5;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 2.1:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.exp(im_m)) + 1.5
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 2.1)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * exp(im_m)) + 1.5);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 2.1)
		tmp = cos(re);
	else
		tmp = (0.5 * exp(im_m)) + 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 2.1], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000009

   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 65.3%

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

   if 2.10000000000000009 < im

   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 100.0%

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

   5. Taylor expanded in re around 0 76.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 76.7%

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

      2. distribute-lft-in 76.7%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 76.7%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 76.7%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot \left(0.25 + im\_m \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.55e+37)
   (cos re)
   (+ 2.0 (* im_m (+ 0.5 (* im_m (+ 0.25 (* im_m 0.08333333333333333))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.55e+37) {
		tmp = cos(re);
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.55d+37) then
        tmp = cos(re)
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * (0.25d0 + (im_m * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.55e+37) {
		tmp = Math.cos(re);
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.55e+37:
		tmp = math.cos(re)
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.55e+37)
		tmp = cos(re);
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.25 + Float64(im_m * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.55e+37)
		tmp = cos(re);
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.55e+37], N[Cos[re], $MachinePrecision], N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.25 + N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5500000000000001e37

   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 62.5%

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

   if 1.5500000000000001e37 < im

   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 100.0%

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

   5. Taylor expanded in re around 0 78.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 78.0%

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

      2. distribute-lft-in 78.0%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 78.0%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 78.0%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 58.8%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 58.8%

         \[\leadsto \color{blue}{im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right) + 2} \]

      2. *-commutative 58.8%

         \[\leadsto im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) + 2 \]

   10. Simplified 58.8%

       \[\leadsto \color{blue}{im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot \left(0.25 + im\_m \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.5)
   1.0
   (+ 2.0 (* im_m (+ 0.5 (* im_m (+ 0.25 (* im_m 0.08333333333333333))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.5) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.5d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * (0.25d0 + (im_m * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.5) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.5:
		tmp = 1.0
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.5)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.25 + Float64(im_m * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.5)
		tmp = 1.0;
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.5], 1.0, N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.25 + N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5

   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 65.3%

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

   4. Taylor expanded in re around 0 35.6%

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

   if 1.5 < im

   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 100.0%

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

   5. Taylor expanded in re around 0 76.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 76.7%

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

      2. distribute-lft-in 76.7%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 76.7%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 76.7%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 51.5%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \color{blue}{im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right) + 2} \]

      2. *-commutative 51.5%

         \[\leadsto im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) + 2 \]

   10. Simplified 51.5%

       \[\leadsto \color{blue}{im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 22.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.2) 1.0 (+ 2.0 (* im_m (+ 0.5 (* im_m 0.25))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.2d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im_m * (0.5d0 + (im_m * 0.25d0)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.2:
		tmp = 1.0
	else:
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.2)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.25))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.2)
		tmp = 1.0;
	else
		tmp = 2.0 + (im_m * (0.5 + (im_m * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.2], 1.0, N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.19999999999999996

   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 65.3%

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

   4. Taylor expanded in re around 0 35.6%

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

   if 1.19999999999999996 < im

   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 100.0%

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

   5. Taylor expanded in re around 0 76.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 76.7%

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

      2. distribute-lft-in 76.7%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 76.7%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 76.7%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 45.8%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.1% accurate, 25.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 310:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 310.0) 1.0 (+ 1.0 (* (* re re) -0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 310.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 310.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 310.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 310.0:
		tmp = 1.0
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 310.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 310.0)
		tmp = 1.0;
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 310.0], 1.0, N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 310

   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 65.0%

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

   4. Taylor expanded in re around 0 35.5%

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

   if 310 < 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 3.1%

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

   4. Taylor expanded in re around 0 12.8%

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

      1. *-commutative 12.8%

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

   6. Simplified 12.8%

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

      1. unpow2 12.8%

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

   8. Applied egg-rr 12.8%

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

Alternative 11: 28.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 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

3. Taylor expanded in im around 0 48.8%

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

4. Taylor expanded in re around 0 26.9%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Alternative 12: 9.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.75 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.75)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.75;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.75d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.75;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.75

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.75
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.75;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.75

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 re around 0 66.6%

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

5. Applied egg-rr 9.0%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]
6. Step-by-step derivation

   1. metadata-eval 9.0%

      \[\leadsto \color{blue}{0.75} \]

7. Applied egg-rr 9.0%

   \[\leadsto \color{blue}{0.75} \]
8. Add Preprocessing

Alternative 13: 7.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.125 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.125)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.125;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.125d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.125;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.125

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.125
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.125;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.125

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 re around 0 66.6%

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

5. Applied egg-rr 7.6%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]
6. Step-by-step derivation

   1. metadata-eval 7.6%

      \[\leadsto \color{blue}{0.125} \]

7. Applied egg-rr 7.6%

   \[\leadsto \color{blue}{0.125} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))