math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 8

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. +-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.3)
   (* (* (cos re) 0.5) (fma im_m im_m 2.0))
   (* (cos re) (* 0.5 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.3) {
		tmp = (cos(re) * 0.5) * fma(im_m, im_m, 2.0);
	} else {
		tmp = cos(re) * (0.5 * exp(im_m));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(cos(re) * Float64(0.5 * exp(im_m)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.2%

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

      1. +-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. fma-define 86.2%

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

   5. Simplified 86.2%

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

   if 1.30000000000000004 < 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. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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

   6. Applied egg-rr 97.5%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   7. Taylor expanded in re around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*r* 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.7:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 0.7) (cos re) (* (cos re) (* 0.5 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.7) {
		tmp = cos(re);
	} else {
		tmp = cos(re) * (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 <= 0.7d0) then
        tmp = cos(re)
    else
        tmp = cos(re) * (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 <= 0.7) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cos(re) * (0.5 * Math.exp(im_m));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 0.7:
		tmp = math.cos(re)
	else:
		tmp = math.cos(re) * (0.5 * math.exp(im_m))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 0.7)
		tmp = cos(re);
	else
		tmp = Float64(cos(re) * 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 <= 0.7)
		tmp = cos(re);
	else
		tmp = cos(re) * (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, 0.7], N[Cos[re], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.2%

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

      1. +-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. fma-define 86.2%

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

   5. Simplified 86.2%

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

   6. Taylor expanded in im around 0 69.6%

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

   if 0.69999999999999996 < 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. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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

   6. Applied egg-rr 97.5%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   7. Taylor expanded in re around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*r* 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.7:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.2) (cos re) (* 0.5 (exp im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.2) {
		tmp = cos(re);
	} else {
		tmp = 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 <= 1.2d0) then
        tmp = cos(re)
    else
        tmp = 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 <= 1.2) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.exp(im_m);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.2:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.exp(im_m)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.2)
		tmp = cos(re);
	else
		tmp = 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 <= 1.2)
		tmp = cos(re);
	else
		tmp = 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, 1.2], N[Cos[re], $MachinePrecision], N[(0.5 * N[Exp[im$95$m], $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 86.2%

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

      1. +-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. fma-define 86.2%

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

   5. Simplified 86.2%

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

   6. Taylor expanded in im around 0 69.6%

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

   if 1.19999999999999996 < 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. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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

   6. Applied egg-rr 97.5%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   7. Taylor expanded in re around 0 73.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 77.5%

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

   1. +-commutative 77.5%

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

   2. unpow2 77.5%

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

   3. fma-define 77.5%

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

5. Simplified 77.5%

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

6. Taylor expanded in im around 0 55.4%

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

7. Final simplification 55.4%

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

Alternative 7: 9.5% accurate, 61.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[(im$95$m + 1.0), $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. +-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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

6. Applied egg-rr 31.5%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

7. Taylor expanded in im around 0 11.5%

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

   1. distribute-lft-out 11.5%

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

   2. distribute-rgt1-in 11.5%

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

9. Simplified 11.5%

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

10. Taylor expanded in re around 0 9.3%

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

    1. +-commutative 9.3%

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

12. Simplified 9.3%

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

13. Final simplification 9.3%

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

Alternative 8: 4.4% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * im$95$m), $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. +-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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

6. Applied egg-rr 31.5%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

7. Taylor expanded in im around 0 11.5%

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

   1. distribute-lft-out 11.5%

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

   2. distribute-rgt1-in 11.5%

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

9. Simplified 11.5%

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

10. Taylor expanded in im around inf 3.2%

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

    1. *-commutative 3.2%

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

12. Simplified 3.2%

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

13. Taylor expanded in re around 0 3.3%

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

14. Final simplification 3.3%

    \[\leadsto 0.5 \cdot im \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))