math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 7

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{e^{re}}\\ t\_0 \cdot \left(\sin im \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (exp re)))) (* t_0 (* (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = sqrt(exp(re));
	return t_0 * (sin(im) * t_0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    t_0 = sqrt(exp(re))
    code = t_0 * (sin(im) * t_0)
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(Math.exp(re));
	return t_0 * (Math.sin(im) * t_0);
}

Python

def code(re, im):
	t_0 = math.sqrt(math.exp(re))
	return t_0 * (math.sin(im) * t_0)

Julia

function code(re, im)
	t_0 = sqrt(exp(re))
	return Float64(t_0 * Float64(sin(im) * t_0))
end

MATLAB

function tmp = code(re, im)
	t_0 = sqrt(exp(re));
	tmp = t_0 * (sin(im) * t_0);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[Exp[re], $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 83.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)}} \]

   2. pow3 83.6%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{re} \cdot \sin im\right)}^{3}}} \]

4. Applied egg-rr 83.6%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{re} \cdot \sin im\right)}^{3}}} \]
5. Step-by-step derivation

   1. rem-cbrt-cube 100.0%

      \[\leadsto \color{blue}{e^{re} \cdot \sin im} \]

   2. *-commutative 100.0%

      \[\leadsto \color{blue}{\sin im \cdot e^{re}} \]

   3. add-sqr-sqrt 100.0%

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

   4. associate-*r* 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \sqrt{e^{re}} \cdot \left(\sin im \cdot \sqrt{e^{re}}\right) \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \sin im \cdot e^{re} \]
4. Add Preprocessing

Alternative 3: 68.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0 68.2%

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

4. Final simplification 68.2%

   \[\leadsto im \cdot e^{re} \]
5. Add Preprocessing

Alternative 4: 51.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 49.0%

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

4. Final simplification 49.0%

   \[\leadsto \sin im \]
5. Add Preprocessing

Alternative 5: 29.5% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + im \cdot re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 29.3%

   \[\leadsto \color{blue}{im + im \cdot re} \]

5. Final simplification 29.3%

   \[\leadsto im + im \cdot re \]
6. Add Preprocessing

Alternative 6: 6.8% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 29.3%

   \[\leadsto \color{blue}{im + im \cdot re} \]

5. Taylor expanded in re around inf 7.8%

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

   1. *-commutative 7.8%

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

7. Simplified 7.8%

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

8. Final simplification 7.8%

   \[\leadsto im \cdot re \]
9. Add Preprocessing

Alternative 7: 26.5% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 25.4%

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

5. Final simplification 25.4%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))