math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 3

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 65.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} + e^{im}\\ \mathbf{if}\;im \leq -5 \cdot 10^{+89} \lor \neg \left(im \leq 2.6 \cdot 10^{+255}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp (- im)) (exp im))))
   (if (or (<= im -5e+89) (not (<= im 2.6e+255)))
     (* t_0 (* 0.5 (+ 1.0 (* -0.5 (* re re)))))
     (* 0.5 t_0))))

C

double code(double re, double im) {
	double t_0 = exp(-im) + exp(im);
	double tmp;
	if ((im <= -5e+89) || !(im <= 2.6e+255)) {
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) + exp(im)
    if ((im <= (-5d+89)) .or. (.not. (im <= 2.6d+255))) then
        tmp = t_0 * (0.5d0 * (1.0d0 + ((-0.5d0) * (re * re))))
    else
        tmp = 0.5d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) + Math.exp(im);
	double tmp;
	if ((im <= -5e+89) || !(im <= 2.6e+255)) {
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) + math.exp(im)
	tmp = 0
	if (im <= -5e+89) or not (im <= 2.6e+255):
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))))
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) + exp(im))
	tmp = 0.0
	if ((im <= -5e+89) || !(im <= 2.6e+255))
		tmp = Float64(t_0 * Float64(0.5 * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) + exp(im);
	tmp = 0.0;
	if ((im <= -5e+89) || ~((im <= 2.6e+255)))
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[im, -5e+89], N[Not[LessEqual[im, 2.6e+255]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < -4.99999999999999983e89 or 2.6000000000000001e255 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 89.7%

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

      1. unpow2 89.7%

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

   4. Simplified 89.7%

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

   if -4.99999999999999983e89 < im < 2.6000000000000001e255

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 63.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+89} \lor \neg \left(im \leq 2.6 \cdot 10^{+255}\right):\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 3: 65.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 63.7%

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

3. Final simplification 63.7%

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

Reproduce

herbie shell --seed 2023195 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))