math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 18

Speedup: 1.0×

• 49.4% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (cos(re) * 0.5) * (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 = (cos(re) * 0.5d0) * (exp(im_m) + exp(-im_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $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. Final simplification 100.0%

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

Alternative 3: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3.5 \lor \neg \left(im\_m \leq 5.6 \cdot 10^{+149}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= im_m 3.5) (not (<= im_m 5.6e+149)))
   (* (* (cos re) 0.5) (fma im_m im_m 2.0))
   (+ 0.5 (* 0.5 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((im_m <= 3.5) || !(im_m <= 5.6e+149)) {
		tmp = (cos(re) * 0.5) * fma(im_m, im_m, 2.0);
	} else {
		tmp = 0.5 + (0.5 * exp(im_m));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((im_m <= 3.5) || !(im_m <= 5.6e+149))
		tmp = Float64(Float64(cos(re) * 0.5) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im_m)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[im$95$m, 3.5], N[Not[LessEqual[im$95$m, 5.6e+149]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.5 or 5.5999999999999998e149 < 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 89.0%

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

      1. +-commutative 89.0%

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

      2. unpow2 89.0%

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

      3. fma-define 89.0%

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

   5. Simplified 89.0%

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

   if 3.5 < im < 5.5999999999999998e149

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 87.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5 \lor \neg \left(im \leq 5.6 \cdot 10^{+149}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.6e-5)
   (cos re)
   (if (<= im_m 5.6e+149)
     (* 0.5 (+ (exp im_m) (exp (- im_m))))
     (* (cos re) (+ 1.0 (* 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 <= 1.6e-5) {
		tmp = cos(re);
	} else if (im_m <= 5.6e+149) {
		tmp = 0.5 * (exp(im_m) + exp(-im_m));
	} else {
		tmp = cos(re) * (1.0 + (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 <= 1.6d-5) then
        tmp = cos(re)
    else if (im_m <= 5.6d+149) then
        tmp = 0.5d0 * (exp(im_m) + exp(-im_m))
    else
        tmp = cos(re) * (1.0d0 + (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 <= 1.6e-5) {
		tmp = Math.cos(re);
	} else if (im_m <= 5.6e+149) {
		tmp = 0.5 * (Math.exp(im_m) + Math.exp(-im_m));
	} else {
		tmp = Math.cos(re) * (1.0 + (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 <= 1.6e-5:
		tmp = math.cos(re)
	elif im_m <= 5.6e+149:
		tmp = 0.5 * (math.exp(im_m) + math.exp(-im_m))
	else:
		tmp = math.cos(re) * (1.0 + (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 <= 1.6e-5)
		tmp = cos(re);
	elseif (im_m <= 5.6e+149)
		tmp = Float64(0.5 * Float64(exp(im_m) + exp(Float64(-im_m))));
	else
		tmp = Float64(cos(re) * Float64(1.0 + 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 <= 1.6e-5)
		tmp = cos(re);
	elseif (im_m <= 5.6e+149)
		tmp = 0.5 * (exp(im_m) + exp(-im_m));
	else
		tmp = cos(re) * (1.0 + (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, 1.6e-5], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 5.6e+149], N[(0.5 * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.59999999999999993e-5

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 70.5%

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

   if 1.59999999999999993e-5 < im < 5.5999999999999998e149

   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 86.5%

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

   if 5.5999999999999998e149 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. *-rgt-identity 94.0%

         \[\leadsto \color{blue}{\cos re \cdot 1} + \left(0.5 \cdot \left(im \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      3. associate-*r* 94.0%

         \[\leadsto \cos re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot im\right) \cdot \cos re} + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      4. associate-*r* 94.0%

         \[\leadsto \cos re \cdot 1 + \left(\left(0.5 \cdot im\right) \cdot \cos re + \color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \cos re}\right) \]

      5. distribute-rgt-out 94.0%

         \[\leadsto \cos re \cdot 1 + \color{blue}{\cos re \cdot \left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      6. +-commutative 94.0%

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      7. distribute-lft-in 94.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      8. +-commutative 94.0%

         \[\leadsto \cos re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)}\right) \]

      9. unpow2 94.0%

         \[\leadsto \cos re \cdot \left(1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

      10. associate-*r* 94.0%

          \[\leadsto \cos re \cdot \left(1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right)\right) \]

      11. distribute-rgt-out 94.0%

          \[\leadsto \cos re \cdot \left(1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)}\right) \]

      12. *-commutative 94.0%

          \[\leadsto \cos re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.25}\right)\right) \]

   8. Simplified 94.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return cos(re) * (0.5 + (0.5 * 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 = cos(re) * (0.5d0 + (0.5d0 * exp(im_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

6. Taylor expanded in re around inf 76.7%

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

   1. *-commutative 76.7%

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

8. Simplified 76.7%

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

9. Final simplification 76.7%

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

Alternative 6: 92.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im\_m}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 2.0)
   (cos re)
   (if (<= im_m 5.6e+149)
     (+ 0.5 (* 0.5 (exp im_m)))
     (* (cos re) (+ 1.0 (* 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 <= 2.0) {
		tmp = cos(re);
	} else if (im_m <= 5.6e+149) {
		tmp = 0.5 + (0.5 * exp(im_m));
	} else {
		tmp = cos(re) * (1.0 + (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 <= 2.0d0) then
        tmp = cos(re)
    else if (im_m <= 5.6d+149) then
        tmp = 0.5d0 + (0.5d0 * exp(im_m))
    else
        tmp = cos(re) * (1.0d0 + (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 <= 2.0) {
		tmp = Math.cos(re);
	} else if (im_m <= 5.6e+149) {
		tmp = 0.5 + (0.5 * Math.exp(im_m));
	} else {
		tmp = Math.cos(re) * (1.0 + (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 <= 2.0:
		tmp = math.cos(re)
	elif im_m <= 5.6e+149:
		tmp = 0.5 + (0.5 * math.exp(im_m))
	else:
		tmp = math.cos(re) * (1.0 + (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 <= 2.0)
		tmp = cos(re);
	elseif (im_m <= 5.6e+149)
		tmp = Float64(0.5 + Float64(0.5 * exp(im_m)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + 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 <= 2.0)
		tmp = cos(re);
	elseif (im_m <= 5.6e+149)
		tmp = 0.5 + (0.5 * exp(im_m));
	else
		tmp = cos(re) * (1.0 + (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, 2.0], N[Cos[re], $MachinePrecision], If[LessEqual[im$95$m, 5.6e+149], N[(0.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 70.3%

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

   if 2 < im < 5.5999999999999998e149

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 87.9%

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

   if 5.5999999999999998e149 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. *-rgt-identity 94.0%

         \[\leadsto \color{blue}{\cos re \cdot 1} + \left(0.5 \cdot \left(im \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      3. associate-*r* 94.0%

         \[\leadsto \cos re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot im\right) \cdot \cos re} + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      4. associate-*r* 94.0%

         \[\leadsto \cos re \cdot 1 + \left(\left(0.5 \cdot im\right) \cdot \cos re + \color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \cos re}\right) \]

      5. distribute-rgt-out 94.0%

         \[\leadsto \cos re \cdot 1 + \color{blue}{\cos re \cdot \left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      6. +-commutative 94.0%

         \[\leadsto \cos re \cdot 1 + \cos re \cdot \color{blue}{\left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      7. distribute-lft-in 94.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      8. +-commutative 94.0%

         \[\leadsto \cos re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)}\right) \]

      9. unpow2 94.0%

         \[\leadsto \cos re \cdot \left(1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

      10. associate-*r* 94.0%

          \[\leadsto \cos re \cdot \left(1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right)\right) \]

      11. distribute-rgt-out 94.0%

          \[\leadsto \cos re \cdot \left(1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)}\right) \]

      12. *-commutative 94.0%

          \[\leadsto \cos re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.25}\right)\right) \]

   8. Simplified 94.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.6e-5) (cos re) (* 0.5 (fma im_m im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.6e-5) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.6e-5)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.6e-5], N[Cos[re], $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.59999999999999993e-5

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 70.5%

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

   if 1.59999999999999993e-5 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 73.9%

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

   4. Taylor expanded in im around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. unpow2 30.5%

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

      3. fma-define 30.5%

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

   6. Simplified 30.5%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 2.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 2.4) (cos re) (+ 0.5 (* 0.5 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 2.4) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (0.5 * exp(im_m));
	}
	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
        tmp = 0.5d0 + (0.5d0 * exp(im_m))
    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 {
		tmp = 0.5 + (0.5 * Math.exp(im_m));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 2.4:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * math.exp(im_m))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 2.4)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im_m)));
	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);
	else
		tmp = 0.5 + (0.5 * exp(im_m));
	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], N[(0.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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. Step-by-step derivation

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 70.3%

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

   if 2.39999999999999991 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 74.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 2.5e+32) (cos re) (+ 1.0 (* 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 <= 2.5e+32) {
		tmp = cos(re);
	} else {
		tmp = 1.0 + (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 <= 2.5d+32) then
        tmp = cos(re)
    else
        tmp = 1.0d0 + (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 <= 2.5e+32) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.0 + (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 <= 2.5e+32:
		tmp = math.cos(re)
	else:
		tmp = 1.0 + (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 <= 2.5e+32)
		tmp = cos(re);
	else
		tmp = Float64(1.0 + 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 <= 2.5e+32)
		tmp = cos(re);
	else
		tmp = 1.0 + (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, 2.5e+32], N[Cos[re], $MachinePrecision], N[(1.0 + 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 < 2.4999999999999999e32

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 67.6%

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

   if 2.4999999999999999e32 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cos-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 74.1%

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

   7. Taylor expanded in im around 0 33.9%

      \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
   8. Step-by-step derivation

      1. +-commutative 33.9%

         \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      2. *-commutative 33.9%

         \[\leadsto 1 + \left(\color{blue}{im \cdot 0.5} + 0.25 \cdot {im}^{2}\right) \]

      3. *-commutative 33.9%

         \[\leadsto 1 + \left(im \cdot 0.5 + \color{blue}{{im}^{2} \cdot 0.25}\right) \]

      4. unpow2 33.9%

         \[\leadsto 1 + \left(im \cdot 0.5 + \color{blue}{\left(im \cdot im\right)} \cdot 0.25\right) \]

      5. associate-*l* 33.9%

         \[\leadsto 1 + \left(im \cdot 0.5 + \color{blue}{im \cdot \left(im \cdot 0.25\right)}\right) \]

      6. distribute-lft-out 33.9%

         \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + im \cdot 0.25\right)} \]

   9. Simplified 33.9%

      \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + im \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.4% accurate, 34.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right) \end{array} \]

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0 + (im_m * (0.5 + (im_m * 0.25)));
}

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 + (im_m * (0.5d0 + (im_m * 0.25d0)))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0 + (im_m * (0.5 + (im_m * 0.25)));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0 + (im_m * (0.5 + (im_m * 0.25)))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.25))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 1.0 + (im_m * (0.5 + (im_m * 0.25)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

6. Taylor expanded in re around 0 48.2%

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

7. Taylor expanded in im around 0 48.5%

   \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
8. Step-by-step derivation

   1. +-commutative 48.5%

      \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

   2. *-commutative 48.5%

      \[\leadsto 1 + \left(\color{blue}{im \cdot 0.5} + 0.25 \cdot {im}^{2}\right) \]

   3. *-commutative 48.5%

      \[\leadsto 1 + \left(im \cdot 0.5 + \color{blue}{{im}^{2} \cdot 0.25}\right) \]

   4. unpow2 48.5%

      \[\leadsto 1 + \left(im \cdot 0.5 + \color{blue}{\left(im \cdot im\right)} \cdot 0.25\right) \]

   5. associate-*l* 48.5%

      \[\leadsto 1 + \left(im \cdot 0.5 + \color{blue}{im \cdot \left(im \cdot 0.25\right)}\right) \]

   6. distribute-lft-out 48.5%

      \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + im \cdot 0.25\right)} \]

9. Simplified 48.5%

   \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + im \cdot 0.25\right)} \]

10. Final simplification 48.5%

    \[\leadsto 1 + im \cdot \left(0.5 + im \cdot 0.25\right) \]
11. Add Preprocessing

Alternative 11: 29.2% accurate, 61.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 + 0.5 \cdot im\_m \end{array} \]

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0 + (0.5 * 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 = 1.0d0 + (0.5d0 * im_m)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0 + (0.5 * im_m);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0 + (0.5 * im_m)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

6. Taylor expanded in re around 0 48.2%

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

7. Taylor expanded in im around 0 31.0%

   \[\leadsto \color{blue}{1 + 0.5 \cdot im} \]

8. Final simplification 31.0%

   \[\leadsto 1 + 0.5 \cdot im \]
9. Add Preprocessing

Alternative 12: 3.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ -2 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 -2.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return -2.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = -2.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return -2.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return -2.0

Julia

im_m = abs(im)
function code(re, im_m)
	return -2.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = -2.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := -2.0

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

7. Applied egg-rr 3.3%

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

8. Final simplification 3.3%

   \[\leadsto -2 \]
9. Add Preprocessing

Alternative 13: 3.7% 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. Step-by-step derivation

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

7. Applied egg-rr 3.7%

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

8. Final simplification 3.7%

   \[\leadsto -1 \]
9. Add Preprocessing

Alternative 14: 7.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.041666666666666664 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.041666666666666664)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.041666666666666664;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.041666666666666664d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.041666666666666664;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.041666666666666664

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.041666666666666664
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.041666666666666664;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.041666666666666664

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

7. Applied egg-rr 7.9%

   \[\leadsto \color{blue}{0.041666666666666664} \]

8. Final simplification 7.9%

   \[\leadsto 0.041666666666666664 \]
9. Add Preprocessing

Alternative 15: 7.8% 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. Step-by-step derivation

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

7. Applied egg-rr 8.3%

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

8. Final simplification 8.3%

   \[\leadsto 0.125 \]
9. Add Preprocessing

Alternative 16: 8.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.25 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.25)

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.25;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.25

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.25
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.25

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

7. Applied egg-rr 8.7%

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

8. Final simplification 8.7%

   \[\leadsto 0.25 \]
9. Add Preprocessing

Alternative 17: 8.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.5)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5;
}

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
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.5;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.5
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.5;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

7. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{0.5} \]

8. Final simplification 9.3%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Alternative 18: 28.6% 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. Step-by-step derivation

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. cos-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 76.7%

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

6. Taylor expanded in re around 0 48.2%

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

7. Taylor expanded in im around 0 31.3%

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

8. Final simplification 31.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))