math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 9

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.000000000005:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   0.0
   (if (<= (exp re) 1.000000000005) (* (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) <= 1.000000000005) {
		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) <= 1.000000000005d0) 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) <= 1.000000000005) {
		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) <= 1.000000000005:
		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) <= 1.000000000005)
		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) <= 1.000000000005)
		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], 1.000000000005], 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. Add Preprocessing

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. +-commutative 55.4%

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

      4. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in im around 0 8.8%

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

       1. associate-+r+ 100.0%

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

       2. distribute-rgt1-in 100.0%

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

       3. metadata-eval 100.0%

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

       4. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if 0.0 < (exp.f64 re) < 1.000000000005

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 99.5%

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

      1. distribute-rgt1-in 99.5%

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

   5. Simplified 99.5%

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

   if 1.000000000005 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 79.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.000000000005:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.000000000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.000000000005)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.000000000005)) {
		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) <= 1.0d0) .or. (.not. (exp(re) <= 1.000000000005d0))) 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) <= 1.0) || !(Math.exp(re) <= 1.000000000005)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.000000000005):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.000000000005))
		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) <= 1.0) || ~((exp(re) <= 1.000000000005)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.000000000005]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.000000000005 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.8%

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

   if 1 < (exp.f64 re) < 1.000000000005

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 95.3%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.000000000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -160:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 70:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -160.0) 0.0 (if (<= re 70.0) (sin im) (pow E re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -160.0) {
		tmp = 0.0;
	} else if (re <= 70.0) {
		tmp = sin(im);
	} else {
		tmp = pow(((double) M_E), re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -160.0) {
		tmp = 0.0;
	} else if (re <= 70.0) {
		tmp = Math.sin(im);
	} else {
		tmp = Math.pow(Math.E, re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -160.0:
		tmp = 0.0
	elif re <= 70.0:
		tmp = math.sin(im)
	else:
		tmp = math.pow(math.e, re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -160.0)
		tmp = 0.0;
	elseif (re <= 70.0)
		tmp = sin(im);
	else
		tmp = exp(1) ^ re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -160.0)
		tmp = 0.0;
	elseif (re <= 70.0)
		tmp = sin(im);
	else
		tmp = 2.71828182845904523536 ^ re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -160.0], 0.0, If[LessEqual[re, 70.0], N[Sin[im], $MachinePrecision], N[Power[E, re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -160

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. +-commutative 55.4%

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

      4. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in im around 0 8.8%

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

       1. associate-+r+ 100.0%

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

       2. distribute-rgt1-in 100.0%

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

       3. metadata-eval 100.0%

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

       4. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if -160 < re < 70

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 97.4%

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

   if 70 < re

   1. Initial program 100.0%

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

      1. add-exp-log 50.0%

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

      2. *-un-lft-identity 50.0%

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

      3. exp-prod 50.0%

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

      4. exp-1-e 50.0%

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

      5. log-prod 50.0%

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

      6. add-log-exp 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in re around inf 50.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -160:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 70:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -130:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -130.0) 0.0 (if (<= re 6.8e-12) (sin im) (+ im (* re im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -130.0) {
		tmp = 0.0;
	} else if (re <= 6.8e-12) {
		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 <= (-130.0d0)) then
        tmp = 0.0d0
    else if (re <= 6.8d-12) 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 <= -130.0) {
		tmp = 0.0;
	} else if (re <= 6.8e-12) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -130.0:
		tmp = 0.0
	elif re <= 6.8e-12:
		tmp = math.sin(im)
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -130.0)
		tmp = 0.0;
	elseif (re <= 6.8e-12)
		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 <= -130.0)
		tmp = 0.0;
	elseif (re <= 6.8e-12)
		tmp = sin(im);
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -130.0], 0.0, If[LessEqual[re, 6.8e-12], N[Sin[im], $MachinePrecision], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -130

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. +-commutative 55.4%

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

      4. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in im around 0 8.8%

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

       1. associate-+r+ 100.0%

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

       2. distribute-rgt1-in 100.0%

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

       3. metadata-eval 100.0%

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

       4. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if -130 < re < 6.8000000000000001e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 97.9%

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

   if 6.8000000000000001e-12 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 79.6%

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

   4. Taylor expanded in re around 0 22.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -130:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.0% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -90:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -90.0) 0.0 (if (<= re 1.0) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -90.0) {
		tmp = 0.0;
	} else 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 <= (-90.0d0)) then
        tmp = 0.0d0
    else 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 <= -90.0) {
		tmp = 0.0;
	} else if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -90.0:
		tmp = 0.0
	elif re <= 1.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -90.0)
		tmp = 0.0;
	elseif (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 <= -90.0)
		tmp = 0.0;
	elseif (re <= 1.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -90.0], 0.0, If[LessEqual[re, 1.0], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -90

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. +-commutative 55.4%

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

      4. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in im around 0 8.8%

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

       1. associate-+r+ 100.0%

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

       2. distribute-rgt1-in 100.0%

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

       3. metadata-eval 100.0%

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

       4. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if -90 < re < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 42.4%

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

   4. Taylor expanded in re around 0 40.7%

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

   if 1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 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 re around inf 4.7%

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

      1. *-commutative 4.7%

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

   8. Simplified 4.7%

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

   9. Taylor expanded in im around 0 20.6%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -90:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.3% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -1.15) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15) {
		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 <= (-1.15d0)) 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 <= -1.15) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15)
		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 <= -1.15)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. +-commutative 55.4%

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

      4. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in im around 0 8.8%

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

       1. associate-+r+ 100.0%

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

       2. distribute-rgt1-in 100.0%

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

       3. metadata-eval 100.0%

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

       4. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if -1.1499999999999999 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 54.5%

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

   4. Taylor expanded in re around 0 35.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.6% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -62:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -62.0) 0.0 im))

C

double code(double re, double im) {
	double tmp;
	if (re <= -62.0) {
		tmp = 0.0;
	} else {
		tmp = 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 <= (-62.0d0)) then
        tmp = 0.0d0
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -62.0) {
		tmp = 0.0;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -62.0:
		tmp = 0.0
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -62.0)
		tmp = 0.0;
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -62.0)
		tmp = 0.0;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -62.0], 0.0, im]

Derivation

1. Split input into 2 regimes

2. if re < -62

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. +-commutative 55.4%

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

      4. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in im around 0 8.8%

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

       1. associate-+r+ 100.0%

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

       2. distribute-rgt1-in 100.0%

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

       3. metadata-eval 100.0%

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

       4. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if -62 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 54.5%

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

   4. Taylor expanded in re around 0 28.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -62:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.8% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 50.8%

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

   1. distribute-rgt1-in 50.8%

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

5. Simplified 50.8%

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

   1. expm1-log1p-u 50.6%

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

   2. expm1-undefine 46.0%

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

   3. log1p-undefine 46.0%

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

   4. rem-exp-log 46.0%

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

7. Applied egg-rr 46.0%

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

   1. sub-neg 46.0%

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

   2. distribute-rgt-in 46.0%

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

   3. +-commutative 46.0%

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

   4. metadata-eval 46.0%

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

9. Applied egg-rr 46.0%

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

10. Taylor expanded in im around 0 4.8%

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

    1. associate-+r+ 29.0%

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

    2. distribute-rgt1-in 29.0%

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

    3. metadata-eval 29.0%

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

    4. mul0-lft 29.0%

       \[\leadsto \color{blue}{0} \]

12. Simplified 29.0%

    \[\leadsto \color{blue}{0} \]

13. Final simplification 29.0%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024062 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))