math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 9

Speedup: 1.0×

• 24.8% 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: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9998) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9998) || !(exp(re) <= 2.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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) <= 0.9998d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(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) <= 0.9998) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9998) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9998) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9998) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99980000000000002 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.8%

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

   if 0.99980000000000002 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

      1. distribute-rgt1-in 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999999999999 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999999999999) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999999999999) || !(exp(re) <= 2.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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) <= 0.9999999999999d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.9999999999999) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999999999999) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999999999999) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999999999999) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999999999999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999999999899969 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 83.1%

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

   if 0.999999999999899969 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999999999999 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{elif}\;re \leq 5200000000000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(re + 1\right) \cdot \left(im \cdot \left(im + 2\right)\right)}{im + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102.0)
   (* (+ re 1.0) 0.0)
   (if (<= re 5200000000000.0)
     (sin im)
     (/ (* (+ re 1.0) (* im (+ im 2.0))) (+ im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102.0) {
		tmp = (re + 1.0) * 0.0;
	} else if (re <= 5200000000000.0) {
		tmp = sin(im);
	} else {
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-102.0d0)) then
        tmp = (re + 1.0d0) * 0.0d0
    else if (re <= 5200000000000.0d0) then
        tmp = sin(im)
    else
        tmp = ((re + 1.0d0) * (im * (im + 2.0d0))) / (im + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -102.0) {
		tmp = (re + 1.0) * 0.0;
	} else if (re <= 5200000000000.0) {
		tmp = Math.sin(im);
	} else {
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102.0:
		tmp = (re + 1.0) * 0.0
	elif re <= 5200000000000.0:
		tmp = math.sin(im)
	else:
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102.0)
		tmp = Float64(Float64(re + 1.0) * 0.0);
	elseif (re <= 5200000000000.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(re + 1.0) * Float64(im * Float64(im + 2.0))) / Float64(im + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -102.0)
		tmp = (re + 1.0) * 0.0;
	elseif (re <= 5200000000000.0)
		tmp = sin(im);
	else
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102.0], N[(N[(re + 1.0), $MachinePrecision] * 0.0), $MachinePrecision], If[LessEqual[re, 5200000000000.0], N[Sin[im], $MachinePrecision], N[(N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -102

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   4. Simplified 2.9%

      \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 2.9%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-udef 50.2%

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

      3. log1p-udef 50.2%

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

      4. rem-exp-log 50.2%

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

   6. Applied egg-rr 50.2%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -102 < re < 5.2e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.0%

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

   if 5.2e12 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.6%

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

      1. distribute-rgt1-in 4.6%

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

   4. Simplified 4.6%

      \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 4.6%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-udef 4.1%

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

      3. log1p-udef 4.1%

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

      4. rem-exp-log 4.1%

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

   6. Applied egg-rr 4.1%

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

   7. Taylor expanded in im around 0 14.5%

      \[\leadsto \left(re + 1\right) \cdot \left(\color{blue}{\left(1 + im\right)} - 1\right) \]
   8. Step-by-step derivation

      1. +-commutative 14.5%

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

   9. Simplified 14.5%

      \[\leadsto \left(re + 1\right) \cdot \left(\color{blue}{\left(im + 1\right)} - 1\right) \]
   10. Step-by-step derivation

       1. flip-- 20.4%

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

       2. associate-*r/ 23.3%

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

       3. +-commutative 23.3%

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

       4. metadata-eval 23.3%

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

       5. difference-of-sqr-1 23.3%

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

       6. associate-+l+ 23.3%

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

       7. metadata-eval 23.3%

          \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + \color{blue}{2}\right) \cdot \left(\left(im + 1\right) - 1\right)\right)}{\left(im + 1\right) + 1} \]

       8. associate--l+ 23.8%

          \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot \color{blue}{\left(im + \left(1 - 1\right)\right)}\right)}{\left(im + 1\right) + 1} \]

       9. metadata-eval 23.8%

          \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot \left(im + \color{blue}{0}\right)\right)}{\left(im + 1\right) + 1} \]

       10. +-rgt-identity 23.8%

           \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot \color{blue}{im}\right)}{\left(im + 1\right) + 1} \]

       11. associate-+l+ 23.8%

           \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot im\right)}{\color{blue}{im + \left(1 + 1\right)}} \]

       12. metadata-eval 23.8%

           \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot im\right)}{im + \color{blue}{2}} \]

   11. Applied egg-rr 23.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{elif}\;re \leq 5200000000000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(re + 1\right) \cdot \left(im \cdot \left(im + 2\right)\right)}{im + 2}\\ \end{array} \]

Alternative 5: 54.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(re + 1\right) \cdot \left(im \cdot \left(im + 2\right)\right)}{im + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   (* (+ re 1.0) 0.0)
   (/ (* (+ re 1.0) (* im (+ im 2.0))) (+ im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = (re + 1.0) * 0.0;
	} else {
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.0d0)) then
        tmp = (re + 1.0d0) * 0.0d0
    else
        tmp = ((re + 1.0d0) * (im * (im + 2.0d0))) / (im + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = (re + 1.0) * 0.0;
	} else {
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = (re + 1.0) * 0.0
	else:
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = Float64(Float64(re + 1.0) * 0.0);
	else
		tmp = Float64(Float64(Float64(re + 1.0) * Float64(im * Float64(im + 2.0))) / Float64(im + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = (re + 1.0) * 0.0;
	else
		tmp = ((re + 1.0) * (im * (im + 2.0))) / (im + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], N[(N[(re + 1.0), $MachinePrecision] * 0.0), $MachinePrecision], N[(N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   4. Simplified 2.9%

      \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 2.9%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-udef 49.4%

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

      3. log1p-udef 49.4%

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

      4. rem-exp-log 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in im around 0 98.3%

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

   if -1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 68.4%

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

      1. distribute-rgt1-in 68.4%

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

   4. Simplified 68.4%

      \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 68.3%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-udef 37.2%

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

      3. log1p-udef 37.2%

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

      4. rem-exp-log 37.2%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in im around 0 8.3%

      \[\leadsto \left(re + 1\right) \cdot \left(\color{blue}{\left(1 + im\right)} - 1\right) \]
   8. Step-by-step derivation

      1. +-commutative 8.3%

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

   9. Simplified 8.3%

      \[\leadsto \left(re + 1\right) \cdot \left(\color{blue}{\left(im + 1\right)} - 1\right) \]
   10. Step-by-step derivation

       1. flip-- 10.1%

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

       2. associate-*r/ 11.0%

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

       3. +-commutative 11.0%

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

       4. metadata-eval 11.0%

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

       5. difference-of-sqr-1 11.0%

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

       6. associate-+l+ 11.0%

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

       7. metadata-eval 11.0%

          \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + \color{blue}{2}\right) \cdot \left(\left(im + 1\right) - 1\right)\right)}{\left(im + 1\right) + 1} \]

       8. associate--l+ 41.8%

          \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot \color{blue}{\left(im + \left(1 - 1\right)\right)}\right)}{\left(im + 1\right) + 1} \]

       9. metadata-eval 41.8%

          \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot \left(im + \color{blue}{0}\right)\right)}{\left(im + 1\right) + 1} \]

       10. +-rgt-identity 41.8%

           \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot \color{blue}{im}\right)}{\left(im + 1\right) + 1} \]

       11. associate-+l+ 41.8%

           \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot im\right)}{\color{blue}{im + \left(1 + 1\right)}} \]

       12. metadata-eval 41.8%

           \[\leadsto \frac{\left(1 + re\right) \cdot \left(\left(im + 2\right) \cdot im\right)}{im + \color{blue}{2}} \]

   11. Applied egg-rr 41.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(re + 1\right) \cdot \left(im \cdot \left(im + 2\right)\right)}{im + 2}\\ \end{array} \]

Alternative 6: 53.2% accurate, 28.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0) (* (+ re 1.0) 0.0) (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = (re + 1.0) * 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 <= (-1.0d0)) then
        tmp = (re + 1.0d0) * 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 <= -1.0) {
		tmp = (re + 1.0) * 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = (re + 1.0) * 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = Float64(Float64(re + 1.0) * 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 <= -1.0)
		tmp = (re + 1.0) * 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   4. Simplified 2.9%

      \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 2.9%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-udef 49.4%

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

      3. log1p-udef 49.4%

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

      4. rem-exp-log 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in im around 0 98.3%

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

   if -1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 56.6%

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

   3. Taylor expanded in re around 0 39.2%

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

4. Final simplification 52.1%

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

Alternative 7: 28.5% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{-25}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.2e-25) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.2e-25) {
		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 <= 1.2d-25) 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 <= 1.2e-25) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.2e-25:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.2e-25)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.2e-25)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.2e-25], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.20000000000000005e-25

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 66.9%

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

   3. Taylor expanded in re around 0 37.6%

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

   if 1.20000000000000005e-25 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 64.0%

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

   3. Taylor expanded in re around 0 14.0%

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

   4. Taylor expanded in re around inf 14.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{-25}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 8: 29.0% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 66.1%

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

3. Taylor expanded in re around 0 31.2%

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

4. Final simplification 31.2%

   \[\leadsto im + re \cdot im \]

Alternative 9: 25.9% 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 66.1%

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

3. Taylor expanded in re around 0 28.1%

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

4. Final simplification 28.1%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023308 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))