math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 10

Speedup: 1.0×

• 23.8% 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. Add Preprocessing

Alternative 2: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.00000025\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.00000025)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.00000025)) {
		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) <= 0.0d0) .or. (.not. (exp(re) <= 1.00000025d0))) 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) <= 0.0) || !(Math.exp(re) <= 1.00000025)) {
		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) <= 0.0) or not (math.exp(re) <= 1.00000025):
		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) <= 0.0) || !(exp(re) <= 1.00000025))
		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) <= 0.0) || ~((exp(re) <= 1.00000025)))
		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], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.00000025]], $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) < 0.0 or 1.00000025000000003 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.9%

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

   if 0.0 < (exp.f64 re) < 1.00000025000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.00000025\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.00000025\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.00000025)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.00000025)) {
		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) <= 0.0d0) .or. (.not. (exp(re) <= 1.00000025d0))) 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) <= 0.0) || !(Math.exp(re) <= 1.00000025)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 1.00000025):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.00000025))
		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) <= 0.0) || ~((exp(re) <= 1.00000025)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.00000025]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.00000025000000003 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.9%

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

   if 0.0 < (exp.f64 re) < 1.00000025000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.00000025\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)\\ \mathbf{if}\;re \leq -8.5 \cdot 10^{+159}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 720:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+166} \lor \neg \left(re \leq 4.2 \cdot 10^{+268}\right) \land re \leq 5.8 \cdot 10^{+284}:\\ \;\;\;\;\frac{t\_0}{im}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{im \cdot \left(1 - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (- im (* re (* re im))))))
   (if (<= re -8.5e+159)
     (* re (/ im re))
     (if (<= re 720.0)
       (sin im)
       (if (or (<= re 3.6e+166) (and (not (<= re 4.2e+268)) (<= re 5.8e+284)))
         (/ t_0 im)
         (/ t_0 (* im (- 1.0 re))))))))

C

double code(double re, double im) {
	double t_0 = im * (im - (re * (re * im)));
	double tmp;
	if (re <= -8.5e+159) {
		tmp = re * (im / re);
	} else if (re <= 720.0) {
		tmp = sin(im);
	} else if ((re <= 3.6e+166) || (!(re <= 4.2e+268) && (re <= 5.8e+284))) {
		tmp = t_0 / im;
	} else {
		tmp = t_0 / (im * (1.0 - re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im - (re * (re * im)))
    if (re <= (-8.5d+159)) then
        tmp = re * (im / re)
    else if (re <= 720.0d0) then
        tmp = sin(im)
    else if ((re <= 3.6d+166) .or. (.not. (re <= 4.2d+268)) .and. (re <= 5.8d+284)) then
        tmp = t_0 / im
    else
        tmp = t_0 / (im * (1.0d0 - re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im - (re * (re * im)));
	double tmp;
	if (re <= -8.5e+159) {
		tmp = re * (im / re);
	} else if (re <= 720.0) {
		tmp = Math.sin(im);
	} else if ((re <= 3.6e+166) || (!(re <= 4.2e+268) && (re <= 5.8e+284))) {
		tmp = t_0 / im;
	} else {
		tmp = t_0 / (im * (1.0 - re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im - (re * (re * im)))
	tmp = 0
	if re <= -8.5e+159:
		tmp = re * (im / re)
	elif re <= 720.0:
		tmp = math.sin(im)
	elif (re <= 3.6e+166) or (not (re <= 4.2e+268) and (re <= 5.8e+284)):
		tmp = t_0 / im
	else:
		tmp = t_0 / (im * (1.0 - re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im - Float64(re * Float64(re * im))))
	tmp = 0.0
	if (re <= -8.5e+159)
		tmp = Float64(re * Float64(im / re));
	elseif (re <= 720.0)
		tmp = sin(im);
	elseif ((re <= 3.6e+166) || (!(re <= 4.2e+268) && (re <= 5.8e+284)))
		tmp = Float64(t_0 / im);
	else
		tmp = Float64(t_0 / Float64(im * Float64(1.0 - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im - (re * (re * im)));
	tmp = 0.0;
	if (re <= -8.5e+159)
		tmp = re * (im / re);
	elseif (re <= 720.0)
		tmp = sin(im);
	elseif ((re <= 3.6e+166) || (~((re <= 4.2e+268)) && (re <= 5.8e+284)))
		tmp = t_0 / im;
	else
		tmp = t_0 / (im * (1.0 - re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im - N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -8.5e+159], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 720.0], N[Sin[im], $MachinePrecision], If[Or[LessEqual[re, 3.6e+166], And[N[Not[LessEqual[re, 4.2e+268]], $MachinePrecision], LessEqual[re, 5.8e+284]]], N[(t$95$0 / im), $MachinePrecision], N[(t$95$0 / N[(im * N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.50000000000000076e159

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.3%

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

      1. distribute-rgt1-in 2.3%

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

   5. Simplified 2.3%

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

   6. Taylor expanded in im around 0 2.2%

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

      1. +-commutative 2.2%

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

   8. Simplified 2.2%

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

   9. Taylor expanded in re around inf 2.2%

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

   10. Taylor expanded in re around 0 37.6%

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

   if -8.50000000000000076e159 < re < 720

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 80.7%

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

   if 720 < re < 3.5999999999999997e166 or 4.2000000000000003e268 < re < 5.7999999999999997e284

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in im around 0 4.3%

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

      1. +-commutative 4.3%

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

   8. Simplified 4.3%

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

      1. distribute-lft-in 4.3%

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

      2. *-rgt-identity 4.3%

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

      3. flip-+ 8.5%

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

      4. unpow2 8.5%

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

   10. Applied egg-rr 8.5%

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

       1. associate-*l* 8.5%

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

       2. unpow2 8.5%

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

       3. distribute-lft-out-- 10.9%

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

       4. sub-neg 10.9%

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

       5. mul-1-neg 10.9%

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

       6. *-commutative 10.9%

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

       7. distribute-lft-out 10.9%

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

   12. Simplified 10.9%

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

   13. Taylor expanded in re around 0 40.8%

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

       1. neg-mul-1 40.8%

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

   15. Simplified 40.8%

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

   if 3.5999999999999997e166 < re < 4.2000000000000003e268 or 5.7999999999999997e284 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.5%

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

      1. distribute-rgt1-in 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in im around 0 28.2%

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

      1. +-commutative 28.2%

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

   8. Simplified 28.2%

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

      1. distribute-lft-in 28.2%

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

      2. *-rgt-identity 28.2%

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

      3. flip-+ 29.9%

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

      4. unpow2 29.9%

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

   10. Applied egg-rr 29.9%

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

       1. associate-*l* 36.8%

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

       2. unpow2 36.8%

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

       3. distribute-lft-out-- 36.8%

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

       4. sub-neg 36.8%

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

       5. mul-1-neg 36.8%

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

       6. *-commutative 36.8%

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

       7. distribute-lft-out 36.8%

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

   12. Simplified 36.8%

       \[\leadsto \color{blue}{\frac{im \cdot \left(re \cdot \left(im \cdot re\right) - im\right)}{im \cdot \left(re + -1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{+159}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 720:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+166} \lor \neg \left(re \leq 4.2 \cdot 10^{+268}\right) \land re \leq 5.8 \cdot 10^{+284}:\\ \;\;\;\;\frac{im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)}{im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)}{im \cdot \left(1 - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)\\ \mathbf{if}\;im \leq 2.6 \cdot 10^{-106}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+49}:\\ \;\;\;\;\frac{t\_0}{im \cdot \left(1 - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (- im (* re (* re im))))))
   (if (<= im 2.6e-106)
     (* re (/ im re))
     (if (<= im 1.55e+49) (/ t_0 (* im (- 1.0 re))) (/ t_0 im)))))

C

double code(double re, double im) {
	double t_0 = im * (im - (re * (re * im)));
	double tmp;
	if (im <= 2.6e-106) {
		tmp = re * (im / re);
	} else if (im <= 1.55e+49) {
		tmp = t_0 / (im * (1.0 - re));
	} else {
		tmp = t_0 / im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im - (re * (re * im)))
    if (im <= 2.6d-106) then
        tmp = re * (im / re)
    else if (im <= 1.55d+49) then
        tmp = t_0 / (im * (1.0d0 - re))
    else
        tmp = t_0 / im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im - (re * (re * im)));
	double tmp;
	if (im <= 2.6e-106) {
		tmp = re * (im / re);
	} else if (im <= 1.55e+49) {
		tmp = t_0 / (im * (1.0 - re));
	} else {
		tmp = t_0 / im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im - (re * (re * im)))
	tmp = 0
	if im <= 2.6e-106:
		tmp = re * (im / re)
	elif im <= 1.55e+49:
		tmp = t_0 / (im * (1.0 - re))
	else:
		tmp = t_0 / im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im - Float64(re * Float64(re * im))))
	tmp = 0.0
	if (im <= 2.6e-106)
		tmp = Float64(re * Float64(im / re));
	elseif (im <= 1.55e+49)
		tmp = Float64(t_0 / Float64(im * Float64(1.0 - re)));
	else
		tmp = Float64(t_0 / im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im - (re * (re * im)));
	tmp = 0.0;
	if (im <= 2.6e-106)
		tmp = re * (im / re);
	elseif (im <= 1.55e+49)
		tmp = t_0 / (im * (1.0 - re));
	else
		tmp = t_0 / im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im - N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.6e-106], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.55e+49], N[(t$95$0 / N[(im * N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.6000000000000001e-106

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 53.6%

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

      1. distribute-rgt1-in 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in im around 0 31.8%

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

      1. +-commutative 31.8%

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

   8. Simplified 31.8%

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

   9. Taylor expanded in re around inf 31.6%

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

   10. Taylor expanded in re around 0 32.5%

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

   if 2.6000000000000001e-106 < im < 1.54999999999999996e49

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 43.2%

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

      1. distribute-rgt1-in 43.2%

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

   5. Simplified 43.2%

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

   6. Taylor expanded in im around 0 34.4%

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

      1. +-commutative 34.4%

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

   8. Simplified 34.4%

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

      1. distribute-lft-in 34.4%

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

      2. *-rgt-identity 34.4%

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

      3. flip-+ 57.9%

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

      4. unpow2 57.9%

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

   10. Applied egg-rr 57.9%

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

       1. associate-*l* 60.8%

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

       2. unpow2 60.8%

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

       3. distribute-lft-out-- 60.8%

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

       4. sub-neg 60.8%

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

       5. mul-1-neg 60.8%

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

       6. *-commutative 60.8%

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

       7. distribute-lft-out 60.8%

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

   12. Simplified 60.8%

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

   if 1.54999999999999996e49 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 56.8%

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

      1. distribute-rgt1-in 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in im around 0 4.1%

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

      1. +-commutative 4.1%

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

   8. Simplified 4.1%

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

      1. distribute-lft-in 4.1%

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

      2. *-rgt-identity 4.1%

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

      3. flip-+ 3.2%

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

      4. unpow2 3.2%

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

   10. Applied egg-rr 3.2%

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

       1. associate-*l* 3.2%

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

       2. unpow2 3.2%

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

       3. distribute-lft-out-- 3.5%

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

       4. sub-neg 3.5%

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

       5. mul-1-neg 3.5%

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

       6. *-commutative 3.5%

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

       7. distribute-lft-out 3.5%

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

   12. Simplified 3.5%

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

   13. Taylor expanded in re around 0 18.0%

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

       1. neg-mul-1 18.0%

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

   15. Simplified 18.0%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-106}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+49}:\\ \;\;\;\;\frac{im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)}{im \cdot \left(1 - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)}{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.2e+34) (* re (/ im re)) (/ (* im (- im (* re (* re im)))) im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.2e+34) {
		tmp = re * (im / re);
	} else {
		tmp = (im * (im - (re * (re * im)))) / 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 <= 1.2d+34) then
        tmp = re * (im / re)
    else
        tmp = (im * (im - (re * (re * im)))) / im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.2e+34) {
		tmp = re * (im / re);
	} else {
		tmp = (im * (im - (re * (re * im)))) / im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.2e+34:
		tmp = re * (im / re)
	else:
		tmp = (im * (im - (re * (re * im)))) / im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.2e+34)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(Float64(im * Float64(im - Float64(re * Float64(re * im)))) / im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.2e+34)
		tmp = re * (im / re);
	else
		tmp = (im * (im - (re * (re * im)))) / im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.2e+34], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im - N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.19999999999999993e34

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 52.0%

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

      1. distribute-rgt1-in 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in im around 0 32.2%

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

      1. +-commutative 32.2%

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

   8. Simplified 32.2%

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

   9. Taylor expanded in re around inf 32.1%

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

   10. Taylor expanded in re around 0 32.6%

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

   if 1.19999999999999993e34 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 56.8%

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

      1. distribute-rgt1-in 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in im around 0 4.1%

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

      1. +-commutative 4.1%

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

   8. Simplified 4.1%

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

      1. distribute-lft-in 4.1%

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

      2. *-rgt-identity 4.1%

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

      3. flip-+ 3.2%

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

      4. unpow2 3.2%

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

   10. Applied egg-rr 3.2%

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

       1. associate-*l* 3.2%

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

       2. unpow2 3.2%

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

       3. distribute-lft-out-- 3.5%

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

       4. sub-neg 3.5%

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

       5. mul-1-neg 3.5%

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

       6. *-commutative 3.5%

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

       7. distribute-lft-out 3.5%

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

   12. Simplified 3.5%

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

   13. Taylor expanded in re around 0 18.0%

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

       1. neg-mul-1 18.0%

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

   15. Simplified 18.0%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(im - re \cdot \left(re \cdot im\right)\right)}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.1% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.8:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.8) (* re (/ im re)) (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.8) {
		tmp = re * (im / re);
	} else {
		tmp = 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 <= (-0.8d0)) then
        tmp = re * (im / re)
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.8) {
		tmp = re * (im / re);
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.8:
		tmp = re * (im / re)
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.8)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.8)
		tmp = re * (im / re);
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.8], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.80000000000000004

   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. Taylor expanded in im around 0 2.6%

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

      1. +-commutative 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in re around inf 2.6%

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

   10. Taylor expanded in re around 0 19.3%

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

   if -0.80000000000000004 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 66.3%

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

      1. distribute-rgt1-in 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in im around 0 33.3%

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

      1. +-commutative 33.3%

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

   8. Simplified 33.3%

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

Alternative 8: 28.1% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7.8e-6) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8e-6) {
		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 <= 7.8d-6) 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 <= 7.8e-6) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8e-6:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8e-6)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8e-6)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8e-6], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.7999999999999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 52.9%

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

      1. distribute-rgt1-in 52.9%

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

   5. Simplified 52.9%

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

   6. Taylor expanded in im around 0 33.6%

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

      1. +-commutative 33.6%

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

   8. Simplified 33.6%

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

   9. Taylor expanded in re around 0 30.0%

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

   if 7.7999999999999999e-6 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 52.8%

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

      1. distribute-rgt1-in 52.8%

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

   5. Simplified 52.8%

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

   6. Taylor expanded in im around 0 4.3%

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

      1. +-commutative 4.3%

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

   8. Simplified 4.3%

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

   9. Taylor expanded in re around inf 4.9%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.5% 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 52.9%

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

   1. distribute-rgt1-in 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in im around 0 26.8%

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

   1. +-commutative 26.8%

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

8. Simplified 26.8%

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

Alternative 10: 26.8% 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 52.9%

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

   1. distribute-rgt1-in 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in im around 0 26.8%

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

   1. +-commutative 26.8%

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

8. Simplified 26.8%

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

9. Taylor expanded in re around 0 23.6%

   \[\leadsto \color{blue}{im} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))