math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 5

Speedup: 1.0×

• 25.0% 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: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+15}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5e+15) (exp re) (* (exp re) (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5e+15) {
		tmp = exp(re);
	} else {
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5d+15) then
        tmp = exp(re)
    else
        tmp = exp(re) * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5e+15) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5e+15:
		tmp = math.exp(re)
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5e+15)
		tmp = exp(re);
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5e+15)
		tmp = exp(re);
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5e+15], N[Exp[re], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5e15

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 61.0%

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

      1. unpow2 61.0%

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

   4. Simplified 61.0%

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

   5. Taylor expanded in im around 0 69.3%

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

   if 5e15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 80.6%

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

      1. unpow2 80.6%

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

   4. Simplified 80.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+15}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 71.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Exp[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 65.8%

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

   1. unpow2 65.8%

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

4. Simplified 65.8%

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

5. Taylor expanded in im around 0 69.7%

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

6. Final simplification 69.7%

   \[\leadsto e^{re} \]

Alternative 4: 12.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -0.5 * (re * (im * im))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 65.8%

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

   1. unpow2 65.8%

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

4. Simplified 65.8%

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

5. Taylor expanded in im around inf 25.4%

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

   1. unpow2 25.4%

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

   2. associate-*r* 25.4%

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

   3. *-commutative 25.4%

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

   4. associate-*l* 25.4%

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

7. Simplified 25.4%

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

8. Taylor expanded in re around 0 13.3%

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

   1. associate-*r* 13.3%

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

   2. distribute-rgt-out 13.4%

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

   3. unpow2 13.4%

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

10. Simplified 13.4%

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

11. Taylor expanded in re around inf 13.7%

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

    1. unpow2 13.7%

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

13. Simplified 13.7%

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

14. Final simplification 13.7%

    \[\leadsto -0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right) \]

Alternative 5: 11.1% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -0.5 * (im * im)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 65.8%

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

   1. unpow2 65.8%

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

4. Simplified 65.8%

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

5. Taylor expanded in im around inf 25.4%

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

   1. unpow2 25.4%

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

   2. associate-*r* 25.4%

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

   3. *-commutative 25.4%

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

   4. associate-*l* 25.4%

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

7. Simplified 25.4%

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

8. Taylor expanded in re around 0 12.2%

   \[\leadsto \color{blue}{-0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 12.2%

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

10. Simplified 12.2%

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

11. Final simplification 12.2%

    \[\leadsto -0.5 \cdot \left(im \cdot im\right) \]

Reproduce

herbie shell --seed 2023193 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))