math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

• 25.3% 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, 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]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.00001:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   0.0
   (if (<= (exp re) 1.00001) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (exp(re) <= 1.00001) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re) * 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) <= 0.0d0) then
        tmp = 0.0d0
    else if (exp(re) <= 1.00001d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (Math.exp(re) <= 1.00001) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = 0.0
	elif math.exp(re) <= 1.00001:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = 0.0;
	elseif (exp(re) <= 1.00001)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = 0.0;
	elseif (exp(re) <= 1.00001)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], 0.0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.00001], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      2. log1p-udef 100.0%

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

      3. rem-exp-log 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right)} - 1 \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right) - 1} \]

   6. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 1.0000100000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.6%

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

      1. distribute-rgt1-in 98.6%

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

   4. Simplified 98.6%

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

   if 1.0000100000000001 < (exp.f64 re)

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 69.2%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.00001:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 3: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.00001:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   0.0
   (if (<= (exp re) 1.00001) (sin im) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (exp(re) <= 1.00001) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * 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) <= 0.0d0) then
        tmp = 0.0d0
    else if (exp(re) <= 1.00001d0) then
        tmp = sin(im)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (Math.exp(re) <= 1.00001) {
		tmp = Math.sin(im);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = 0.0
	elif math.exp(re) <= 1.00001:
		tmp = math.sin(im)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = 0.0;
	elseif (exp(re) <= 1.00001)
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = 0.0;
	elseif (exp(re) <= 1.00001)
		tmp = sin(im);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], 0.0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.00001], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      2. log1p-udef 100.0%

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

      3. rem-exp-log 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right)} - 1 \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right) - 1} \]

   6. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 1.0000100000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.0%

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

   if 1.0000100000000001 < (exp.f64 re)

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 69.2%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.00001:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 4: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -88:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -88.0) 0.0 (if (<= re 2.2e-8) (sin im) (+ im (* re im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -88.0) {
		tmp = 0.0;
	} else if (re <= 2.2e-8) {
		tmp = sin(im);
	} else {
		tmp = im + (re * 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 <= (-88.0d0)) then
        tmp = 0.0d0
    else if (re <= 2.2d-8) then
        tmp = sin(im)
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -88.0) {
		tmp = 0.0;
	} else if (re <= 2.2e-8) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -88.0:
		tmp = 0.0
	elif re <= 2.2e-8:
		tmp = math.sin(im)
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -88.0)
		tmp = 0.0;
	elseif (re <= 2.2e-8)
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -88.0)
		tmp = 0.0;
	elseif (re <= 2.2e-8)
		tmp = sin(im);
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -88.0], 0.0, If[LessEqual[re, 2.2e-8], N[Sin[im], $MachinePrecision], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -88

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      2. log1p-udef 100.0%

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

      3. rem-exp-log 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right)} - 1 \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right) - 1} \]

   6. Taylor expanded in im around 0 100.0%

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

   if -88 < re < 2.1999999999999998e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.0%

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

   if 2.1999999999999998e-8 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 69.2%

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

   3. Taylor expanded in re around 0 15.4%

      \[\leadsto \color{blue}{im + im \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -88:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 5: 53.3% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.85:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.0255:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.85) 0.0 (if (<= re 0.0255) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.85) {
		tmp = 0.0;
	} else if (re <= 0.0255) {
		tmp = im;
	} else {
		tmp = re * 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 <= (-0.85d0)) then
        tmp = 0.0d0
    else if (re <= 0.0255d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.85) {
		tmp = 0.0;
	} else if (re <= 0.0255) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.85:
		tmp = 0.0
	elif re <= 0.0255:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.85)
		tmp = 0.0;
	elseif (re <= 0.0255)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.85)
		tmp = 0.0;
	elseif (re <= 0.0255)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.85], 0.0, If[LessEqual[re, 0.0255], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.849999999999999978

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      2. log1p-udef 100.0%

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

      3. rem-exp-log 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right)} - 1 \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right) - 1} \]

   6. Taylor expanded in im around 0 98.4%

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

   if -0.849999999999999978 < re < 0.0254999999999999984

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 52.0%

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

   3. Taylor expanded in re around 0 51.5%

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

   if 0.0254999999999999984 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.7%

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

   3. Taylor expanded in re around 0 14.6%

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

   4. Taylor expanded in re around inf 14.6%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.85:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.0255:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 6: 53.4% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.85:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -0.85) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.85) {
		tmp = 0.0;
	} else {
		tmp = im + (re * 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 <= (-0.85d0)) then
        tmp = 0.0d0
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.85) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.85:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.85)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.85)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.85], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.849999999999999978

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      2. log1p-udef 100.0%

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

      3. rem-exp-log 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right)} - 1 \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{re}, \sin im, 1\right) - 1} \]

   6. Taylor expanded in im around 0 98.4%

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

   if -0.849999999999999978 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 57.8%

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

   3. Taylor expanded in re around 0 38.9%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.85:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 7: 29.4% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.0255:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 0.0255) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.0255) {
		tmp = im;
	} else {
		tmp = re * 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 <= 0.0255d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.0255) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.0255:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.0255)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.0255)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.0255], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.0254999999999999984

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 67.3%

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

   3. Taylor expanded in re around 0 35.8%

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

   if 0.0254999999999999984 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.7%

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

   3. Taylor expanded in re around 0 14.6%

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

   4. Taylor expanded in re around inf 14.6%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.0255:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 8: 26.1% 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. Taylor expanded in im around 0 67.6%

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

3. Taylor expanded in re around 0 27.2%

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

4. Final simplification 27.2%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023320 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))