math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 8

Speedup: 1.0×

• 24.1% 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.5) (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) <= 0.5) || !(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) <= 0.5d0) .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) <= 0.5) || !(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) <= 0.5) 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) <= 0.5) || !(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) <= 0.5) || ~((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], 0.5], 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) < 0.5 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.4%

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

   if 0.5 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.5) (not (<= (exp re) 1.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.5) || !(exp(re) <= 1.0)) {
		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) <= 0.5d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.5) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.5) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.5) || !(exp(re) <= 1.0))
		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) <= 0.5) || ~((exp(re) <= 1.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.5], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.5 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.0%

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

   if 0.5 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.0118:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.0118) (sin im) (/ (- (/ re (/ (/ 1.0 re) im)) im) (+ re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.0118) {
		tmp = sin(im);
	} else {
		tmp = ((re / ((1.0 / re) / im)) - 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 (re <= 0.0118d0) then
        tmp = sin(im)
    else
        tmp = ((re / ((1.0d0 / re) / im)) - im) / (re + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.0118) {
		tmp = Math.sin(im);
	} else {
		tmp = ((re / ((1.0 / re) / im)) - im) / (re + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.0118:
		tmp = math.sin(im)
	else:
		tmp = ((re / ((1.0 / re) / im)) - im) / (re + -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.0118)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(re / Float64(Float64(1.0 / re) / im)) - im) / Float64(re + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.0118)
		tmp = sin(im);
	else
		tmp = ((re / ((1.0 / re) / im)) - im) / (re + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.0118], N[Sin[im], $MachinePrecision], N[(N[(N[(re / N[(N[(1.0 / re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] / N[(re + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.0117999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 62.9%

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

   if 0.0117999999999999997 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.7%

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

      1. distribute-rgt1-in 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in im around 0 9.0%

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

      1. +-commutative 9.0%

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

   8. Simplified 9.0%

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

      1. distribute-lft-in 9.0%

         \[\leadsto \color{blue}{im \cdot re + im \cdot 1} \]

      2. *-rgt-identity 9.0%

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

   10. Applied egg-rr 9.0%

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

       1. flip-+ 8.2%

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

       2. div-sub 8.2%

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

       3. pow2 8.2%

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

       4. *-commutative 8.2%

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

       5. *-un-lft-identity 8.2%

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

       6. distribute-rgt-out-- 8.2%

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

       7. pow2 8.2%

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

       8. *-commutative 8.2%

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

       9. *-un-lft-identity 8.2%

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

       10. distribute-rgt-out-- 8.2%

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

   12. Applied egg-rr 8.2%

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

       1. div-sub 8.2%

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

       2. associate-/r* 14.4%

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

       3. div-sub 14.4%

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

       4. unpow2 14.4%

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

       5. associate-/l* 16.3%

          \[\leadsto \frac{\color{blue}{\frac{im \cdot re}{\frac{im}{im \cdot re}}} - \frac{{im}^{2}}{im}}{re - 1} \]

       6. *-commutative 16.3%

          \[\leadsto \frac{\frac{\color{blue}{re \cdot im}}{\frac{im}{im \cdot re}} - \frac{{im}^{2}}{im}}{re - 1} \]

       7. associate-/l* 16.3%

          \[\leadsto \frac{\color{blue}{\frac{re}{\frac{\frac{im}{im \cdot re}}{im}}} - \frac{{im}^{2}}{im}}{re - 1} \]

       8. associate-/r* 16.3%

          \[\leadsto \frac{\frac{re}{\frac{\color{blue}{\frac{\frac{im}{im}}{re}}}{im}} - \frac{{im}^{2}}{im}}{re - 1} \]

       9. *-inverses 16.3%

          \[\leadsto \frac{\frac{re}{\frac{\frac{\color{blue}{1}}{re}}{im}} - \frac{{im}^{2}}{im}}{re - 1} \]

       10. unpow2 16.3%

           \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \frac{\color{blue}{im \cdot im}}{im}}{re - 1} \]

       11. associate-/l* 19.6%

           \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{\frac{im}{\frac{im}{im}}}}{re - 1} \]

       12. associate-/r/ 19.6%

           \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{\frac{im}{im} \cdot im}}{re - 1} \]

       13. *-inverses 19.6%

           \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{1} \cdot im}{re - 1} \]

       14. *-lft-identity 19.6%

           \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{im}}{re - 1} \]

       15. sub-neg 19.6%

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

       16. metadata-eval 19.6%

           \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + \color{blue}{-1}} \]

   14. Simplified 19.6%

       \[\leadsto \color{blue}{\frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.0118:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.6% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + -1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (- (/ re (/ (/ 1.0 re) im)) im) (+ re -1.0)))

C

double code(double re, double im) {
	return ((re / ((1.0 / re) / im)) - im) / (re + -1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((re / ((1.0d0 / re) / im)) - im) / (re + (-1.0d0))
end function

Java

public static double code(double re, double im) {
	return ((re / ((1.0 / re) / im)) - im) / (re + -1.0);
}

Python

def code(re, im):
	return ((re / ((1.0 / re) / im)) - im) / (re + -1.0)

Julia

function code(re, im)
	return Float64(Float64(Float64(re / Float64(Float64(1.0 / re) / im)) - im) / Float64(re + -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = ((re / ((1.0 / re) / im)) - im) / (re + -1.0);
end

Wolfram

code[re_, im_] := N[(N[(N[(re / N[(N[(1.0 / re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] / N[(re + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 48.6%

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

   1. distribute-rgt1-in 48.6%

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

5. Simplified 48.6%

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

6. Taylor expanded in im around 0 28.4%

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

   1. +-commutative 28.4%

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

8. Simplified 28.4%

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

   1. distribute-lft-in 28.4%

      \[\leadsto \color{blue}{im \cdot re + im \cdot 1} \]

   2. *-rgt-identity 28.4%

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

10. Applied egg-rr 28.4%

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

    1. flip-+ 17.1%

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

    2. div-sub 17.1%

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

    3. pow2 17.1%

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

    4. *-commutative 17.1%

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

    5. *-un-lft-identity 17.1%

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

    6. distribute-rgt-out-- 17.1%

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

    7. pow2 17.1%

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

    8. *-commutative 17.1%

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

    9. *-un-lft-identity 17.1%

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

    10. distribute-rgt-out-- 17.1%

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

12. Applied egg-rr 17.1%

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

    1. div-sub 17.1%

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

    2. associate-/r* 18.6%

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

    3. div-sub 18.6%

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

    4. unpow2 18.6%

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

    5. associate-/l* 18.4%

       \[\leadsto \frac{\color{blue}{\frac{im \cdot re}{\frac{im}{im \cdot re}}} - \frac{{im}^{2}}{im}}{re - 1} \]

    6. *-commutative 18.4%

       \[\leadsto \frac{\frac{\color{blue}{re \cdot im}}{\frac{im}{im \cdot re}} - \frac{{im}^{2}}{im}}{re - 1} \]

    7. associate-/l* 18.4%

       \[\leadsto \frac{\color{blue}{\frac{re}{\frac{\frac{im}{im \cdot re}}{im}}} - \frac{{im}^{2}}{im}}{re - 1} \]

    8. associate-/r* 18.4%

       \[\leadsto \frac{\frac{re}{\frac{\color{blue}{\frac{\frac{im}{im}}{re}}}{im}} - \frac{{im}^{2}}{im}}{re - 1} \]

    9. *-inverses 18.4%

       \[\leadsto \frac{\frac{re}{\frac{\frac{\color{blue}{1}}{re}}{im}} - \frac{{im}^{2}}{im}}{re - 1} \]

    10. unpow2 18.4%

        \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \frac{\color{blue}{im \cdot im}}{im}}{re - 1} \]

    11. associate-/l* 31.0%

        \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{\frac{im}{\frac{im}{im}}}}{re - 1} \]

    12. associate-/r/ 31.0%

        \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{\frac{im}{im} \cdot im}}{re - 1} \]

    13. *-inverses 31.0%

        \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{1} \cdot im}{re - 1} \]

    14. *-lft-identity 31.0%

        \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - \color{blue}{im}}{re - 1} \]

    15. sub-neg 31.0%

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

    16. metadata-eval 31.0%

        \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + \color{blue}{-1}} \]

14. Simplified 31.0%

    \[\leadsto \color{blue}{\frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + -1}} \]

15. Final simplification 31.0%

    \[\leadsto \frac{\frac{re}{\frac{\frac{1}{re}}{im}} - im}{re + -1} \]
16. Add Preprocessing

Alternative 6: 28.8% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+46}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3.4e+46) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.4e+46) {
		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 (im <= 3.4d+46) 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 (im <= 3.4e+46) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.4e+46:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.4e+46)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.4e+46)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.4e+46], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.3999999999999998e46

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 48.4%

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

      1. distribute-rgt1-in 48.4%

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

   5. Simplified 48.4%

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

   6. Taylor expanded in im around 0 37.5%

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

      1. +-commutative 37.5%

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

   8. Simplified 37.5%

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

   9. Taylor expanded in re around 0 36.1%

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

   if 3.3999999999999998e46 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 49.4%

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

      1. distribute-rgt1-in 49.4%

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

   5. Simplified 49.4%

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

   6. Taylor expanded in im around 0 3.3%

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

      1. +-commutative 3.3%

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

   8. Simplified 3.3%

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

   9. Taylor expanded in re around inf 4.3%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+46}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.2% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ re 1.0)))

C

double code(double re, double im) {
	return im * (re + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (re + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return im * (re + 1.0);
}

Python

def code(re, im):
	return im * (re + 1.0)

Julia

function code(re, im)
	return Float64(im * Float64(re + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (re + 1.0);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 48.6%

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

   1. distribute-rgt1-in 48.6%

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

5. Simplified 48.6%

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

6. Taylor expanded in im around 0 28.4%

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

   1. +-commutative 28.4%

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

8. Simplified 28.4%

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

9. Final simplification 28.4%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 8: 27.3% 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 re around 0 48.6%

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

   1. distribute-rgt1-in 48.6%

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

5. Simplified 48.6%

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

6. Taylor expanded in im around 0 28.4%

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

   1. +-commutative 28.4%

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

8. Simplified 28.4%

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

9. Taylor expanded in re around 0 27.0%

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

10. Final simplification 27.0%

    \[\leadsto im \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))