math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 6

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]

Alternative 2: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 2.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 74.7%

      \[\leadsto \color{blue}{e^{re}} \]

   if 1 < (exp.f64 re) < 2

   1. Initial program 99.2%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 84.6%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   3. Step-by-step derivation

      1. distribute-rgt1-in 84.6%

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

   4. Simplified 84.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3: 71.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 74.7%

      \[\leadsto \color{blue}{e^{re}} \]

   if 1 < (exp.f64 re) < 2

   1. Initial program 99.2%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 50.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]

Alternative 4: 50.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[im], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0 47.9%

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

3. Final simplification 47.9%

   \[\leadsto \cos im \]

Alternative 5: 29.0% accurate, 67.7× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

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

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0 48.6%

   \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
3. Step-by-step derivation

   1. distribute-rgt1-in 48.6%

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

4. Simplified 48.6%

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

5. Taylor expanded in im around 0 29.3%

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

6. Final simplification 29.3%

   \[\leadsto re + 1 \]

Alternative 6: 28.5% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0 47.9%

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

3. Taylor expanded in im around 0 29.0%

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

4. Final simplification 29.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023298 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))