math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.998) (not (<= (exp re) 1.1)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.998) || !(exp(re) <= 1.1)) {
		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) <= 0.998d0) .or. (.not. (exp(re) <= 1.1d0))) 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) <= 0.998) || !(Math.exp(re) <= 1.1)) {
		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) <= 0.998) or not (math.exp(re) <= 1.1):
		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) <= 0.998) || !(exp(re) <= 1.1))
		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) <= 0.998) || ~((exp(re) <= 1.1)))
		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], 0.998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.1]], $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) < 0.998 or 1.1000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.5%

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

   if 0.998 < (exp.f64 re) < 1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.8%

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

      1. distribute-rgt1-in 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.998) (not (<= (exp re) 1.1)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.998) || !(exp(re) <= 1.1)) {
		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) <= 0.998d0) .or. (.not. (exp(re) <= 1.1d0))) 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) <= 0.998) || !(Math.exp(re) <= 1.1)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.998) or not (math.exp(re) <= 1.1):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.998) || !(exp(re) <= 1.1))
		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) <= 0.998) || ~((exp(re) <= 1.1)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.1]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.998 or 1.1000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.5%

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

   if 0.998 < (exp.f64 re) < 1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+116}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(re \cdot re\right) - im}{re + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.35e+116) (sin im) (/ (- (* im (* re re)) im) (+ re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.35e+116) {
		tmp = sin(im);
	} else {
		tmp = ((im * (re * re)) - 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 <= 1.35d+116) then
        tmp = sin(im)
    else
        tmp = ((im * (re * re)) - im) / (re + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.35e+116) {
		tmp = Math.sin(im);
	} else {
		tmp = ((im * (re * re)) - im) / (re + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.35e+116:
		tmp = math.sin(im)
	else:
		tmp = ((im * (re * re)) - im) / (re + -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.35e+116)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(im * Float64(re * re)) - im) / Float64(re + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.35e+116)
		tmp = sin(im);
	else
		tmp = ((im * (re * re)) - im) / (re + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.35e+116], N[Sin[im], $MachinePrecision], N[(N[(N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] / N[(re + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.35e116

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 62.0%

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

   if 1.35e116 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.7%

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

      1. distribute-rgt1-in 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in im around 0 14.2%

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

      1. +-commutative 14.2%

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

      2. distribute-rgt-in 14.2%

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

      3. *-un-lft-identity 14.2%

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

   8. Applied egg-rr 14.2%

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

      1. flip-+ 13.0%

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

      2. div-sub 13.0%

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

      3. pow2 13.0%

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

      4. *-un-lft-identity 13.0%

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

      5. distribute-rgt-out-- 13.0%

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

      6. sub-neg 13.0%

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

      7. metadata-eval 13.0%

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

      8. pow2 13.0%

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

      9. *-un-lft-identity 13.0%

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

      10. distribute-rgt-out-- 13.0%

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

      11. sub-neg 13.0%

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

      12. metadata-eval 13.0%

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

   10. Applied egg-rr 13.0%

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

       1. div-sub 13.0%

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

       2. associate-/r* 19.8%

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

       3. div-sub 19.8%

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

       4. unpow2 19.8%

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

       5. *-commutative 19.8%

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

       6. *-commutative 19.8%

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

       7. swap-sqr 23.7%

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

       8. unpow2 23.7%

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

       9. *-rgt-identity 23.7%

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

       10. times-frac 23.7%

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

       11. unpow2 23.7%

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

       12. associate-/l* 37.7%

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

       13. *-inverses 37.7%

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

       14. *-rgt-identity 37.7%

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

       15. unpow2 37.7%

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

       16. associate-/l* 49.4%

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

       17. *-inverses 49.4%

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

       18. *-rgt-identity 49.4%

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

   12. Simplified 49.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+116}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(re \cdot re\right) - im}{re + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.3% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[(im * N[(re * re), $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 52.5%

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

   1. distribute-rgt1-in 52.5%

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

5. Simplified 52.5%

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

6. Taylor expanded in im around 0 25.9%

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

   1. +-commutative 25.9%

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

   2. distribute-rgt-in 25.9%

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

   3. *-un-lft-identity 25.9%

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

8. Applied egg-rr 25.9%

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

   1. flip-+ 14.7%

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

   2. div-sub 14.7%

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

   3. pow2 14.7%

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

   4. *-un-lft-identity 14.7%

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

   5. distribute-rgt-out-- 14.7%

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

   6. sub-neg 14.7%

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

   7. metadata-eval 14.7%

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

   8. pow2 14.7%

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

   9. *-un-lft-identity 14.7%

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

   10. distribute-rgt-out-- 14.7%

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

   11. sub-neg 14.7%

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

   12. metadata-eval 14.7%

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

10. Applied egg-rr 14.7%

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

    1. div-sub 14.7%

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

    2. associate-/r* 15.8%

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

    3. div-sub 15.8%

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

    4. unpow2 15.8%

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

    5. *-commutative 15.8%

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

    6. *-commutative 15.8%

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

    7. swap-sqr 17.7%

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

    8. unpow2 17.7%

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

    9. *-rgt-identity 17.7%

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

    10. times-frac 17.0%

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

    11. unpow2 17.0%

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

    12. associate-/l* 17.1%

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

    13. *-inverses 17.1%

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

    14. *-rgt-identity 17.1%

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

    15. unpow2 17.1%

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

    16. associate-/l* 31.7%

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

    17. *-inverses 31.7%

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

    18. *-rgt-identity 31.7%

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

12. Simplified 31.7%

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

13. Final simplification 31.7%

    \[\leadsto \frac{im \cdot \left(re \cdot re\right) - im}{re + -1} \]
14. Add Preprocessing

Alternative 6: 30.2% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 68.3%

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

      1. distribute-rgt1-in 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in im around 0 31.1%

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

   7. Taylor expanded in re around 0 31.3%

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

   if 1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.0%

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

      1. distribute-rgt1-in 5.0%

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

   5. Simplified 5.0%

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

   6. Taylor expanded in im around 0 10.4%

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

   7. Taylor expanded in re around inf 10.4%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
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 52.5%

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

   1. distribute-rgt1-in 52.5%

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

5. Simplified 52.5%

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

6. Taylor expanded in im around 0 25.9%

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

7. Final simplification 25.9%

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

Alternative 8: 26.9% 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. Add Preprocessing

3. Taylor expanded in re around 0 52.5%

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

   1. distribute-rgt1-in 52.5%

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

5. Simplified 52.5%

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

6. Taylor expanded in im around 0 25.9%

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

7. Taylor expanded in re around 0 24.1%

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

8. Final simplification 24.1%

   \[\leadsto im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))