math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 4

Speedup: 1.0×

• 23.5% 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: 71.4% 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 63.5%

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

   1. unpow2 63.5%

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

4. Simplified 63.5%

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

5. Taylor expanded in im around 0 73.4%

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

6. Final simplification 73.4%

   \[\leadsto e^{re} \]

Alternative 3: 12.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 63.5%

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

   1. unpow2 63.5%

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

4. Simplified 63.5%

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

5. Taylor expanded in im around inf 27.1%

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

   1. *-commutative 27.1%

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

   2. associate-*r* 27.1%

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

   3. unpow2 27.1%

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

   4. associate-*l* 31.4%

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

   5. *-commutative 31.4%

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

7. Simplified 31.4%

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

8. Taylor expanded in re around 0 8.8%

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

   1. distribute-lft-out 8.8%

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

10. Simplified 8.8%

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

11. Taylor expanded in re around inf 12.3%

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

    1. *-commutative 12.3%

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

    2. unpow2 12.3%

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

    3. associate-*l* 12.3%

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

    4. *-commutative 12.3%

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

13. Simplified 12.3%

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

14. Final simplification 12.3%

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

Alternative 4: 10.9% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 63.5%

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

   1. unpow2 63.5%

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

4. Simplified 63.5%

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

5. Taylor expanded in im around inf 27.1%

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

   1. *-commutative 27.1%

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

   2. associate-*r* 27.1%

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

   3. unpow2 27.1%

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

   4. associate-*l* 31.4%

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

   5. *-commutative 31.4%

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

7. Simplified 31.4%

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

8. Taylor expanded in re around 0 11.5%

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

9. Final simplification 11.5%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))