math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 15

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999995) (not (<= (exp re) 1.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999995) || !(exp(re) <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.999995d0) .or. (.not. (exp(re) <= 1.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.999995) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.999995) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.999995) || !(exp(re) <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.999995) || ~((exp(re) <= 1.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999499999999997 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.3%

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

   if 0.99999499999999997 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.044 \lor \neg \left(re \leq 0.076\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.044) (and (not (<= re 0.076)) (<= re 1.05e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ (+ re 1.0) (* (* re re) (+ (* re 0.16666666666666666) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.044) || (!(re <= 0.076) && (re <= 1.05e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.044d0)) .or. (.not. (re <= 0.076d0)) .and. (re <= 1.05d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.044) || (!(re <= 0.076) && (re <= 1.05e+103))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.044) or (not (re <= 0.076) and (re <= 1.05e+103)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.044) || (!(re <= 0.076) && (re <= 1.05e+103)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.044) || (~((re <= 0.076)) && (re <= 1.05e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.044], And[N[Not[LessEqual[re, 0.076]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.043999999999999997 or 0.0759999999999999981 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.6%

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

   if -0.043999999999999997 < re < 0.0759999999999999981 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + \left(0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.6%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + \left(0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]

      2. *-commutative 99.6%

         \[\leadsto \left(\sin im + \color{blue}{re \cdot \sin im}\right) + \left(0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right) \]

      3. distribute-rgt1-in 99.6%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + \left(0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right) \]

      4. *-commutative 99.6%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} + \left(0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right) \]

      5. +-commutative 99.6%

         \[\leadsto \sin im \cdot \left(re + 1\right) + \color{blue}{\left(0.5 \cdot \left(\sin im \cdot {re}^{2}\right) + 0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right)\right)} \]

      6. *-commutative 99.6%

         \[\leadsto \sin im \cdot \left(re + 1\right) + \left(0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} + 0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right)\right) \]

      7. associate-*r* 99.6%

         \[\leadsto \sin im \cdot \left(re + 1\right) + \left(\color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} + 0.16666666666666666 \cdot \left(\sin im \cdot {re}^{3}\right)\right) \]

      8. *-commutative 99.6%

         \[\leadsto \sin im \cdot \left(re + 1\right) + \left(\left(0.5 \cdot {re}^{2}\right) \cdot \sin im + 0.16666666666666666 \cdot \color{blue}{\left({re}^{3} \cdot \sin im\right)}\right) \]

      9. associate-*r* 99.6%

         \[\leadsto \sin im \cdot \left(re + 1\right) + \left(\left(0.5 \cdot {re}^{2}\right) \cdot \sin im + \color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im}\right) \]

      10. distribute-rgt-out 99.6%

          \[\leadsto \sin im \cdot \left(re + 1\right) + \color{blue}{\sin im \cdot \left(0.5 \cdot {re}^{2} + 0.16666666666666666 \cdot {re}^{3}\right)} \]

      11. distribute-lft-out 99.6%

          \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + \left(0.5 \cdot {re}^{2} + 0.16666666666666666 \cdot {re}^{3}\right)\right)} \]

      12. +-commutative 99.6%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)}\right) \]

   4. Simplified 99.6%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.044 \lor \neg \left(re \leq 0.076\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \]

Alternative 4: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.043:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 0.015:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (* (exp re) im)))
   (if (<= re -0.043)
     t_1
     (if (<= re 0.015)
       (* (sin im) (+ (+ re 1.0) t_0))
       (if (<= re 8.8e+138) t_1 (* (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = exp(re) * im;
	double tmp;
	if (re <= -0.043) {
		tmp = t_1;
	} else if (re <= 0.015) {
		tmp = sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 8.8e+138) {
		tmp = t_1;
	} else {
		tmp = sin(im) * t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = exp(re) * im
    if (re <= (-0.043d0)) then
        tmp = t_1
    else if (re <= 0.015d0) then
        tmp = sin(im) * ((re + 1.0d0) + t_0)
    else if (re <= 8.8d+138) then
        tmp = t_1
    else
        tmp = sin(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.043) {
		tmp = t_1;
	} else if (re <= 0.015) {
		tmp = Math.sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 8.8e+138) {
		tmp = t_1;
	} else {
		tmp = Math.sin(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = math.exp(re) * im
	tmp = 0
	if re <= -0.043:
		tmp = t_1
	elif re <= 0.015:
		tmp = math.sin(im) * ((re + 1.0) + t_0)
	elif re <= 8.8e+138:
		tmp = t_1
	else:
		tmp = math.sin(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.043)
		tmp = t_1;
	elseif (re <= 0.015)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 8.8e+138)
		tmp = t_1;
	else
		tmp = Float64(sin(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.043)
		tmp = t_1;
	elseif (re <= 0.015)
		tmp = sin(im) * ((re + 1.0) + t_0);
	elseif (re <= 8.8e+138)
		tmp = t_1;
	else
		tmp = sin(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.043], t$95$1, If[LessEqual[re, 0.015], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.8e+138], t$95$1, N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.042999999999999997 or 0.014999999999999999 < re < 8.8000000000000003e138

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

   if -0.042999999999999997 < re < 0.014999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 99.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 99.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 99.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 99.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 99.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 99.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 99.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 99.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 99.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 99.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   if 8.8000000000000003e138 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 97.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 97.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 97.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 97.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 97.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 97.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 97.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 97.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 97.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 97.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 97.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 97.0%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 97.0%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 97.0%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 97.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.043:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.015:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 95.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0028:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0028)
     t_0
     (if (<= re 8.7e-16)
       (* (sin im) (+ re 1.0))
       (if (<= re 8.8e+138) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0028) {
		tmp = t_0;
	} else if (re <= 8.7e-16) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 8.8e+138) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	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 = exp(re) * im
    if (re <= (-0.0028d0)) then
        tmp = t_0
    else if (re <= 8.7d-16) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 8.8d+138) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.0028) {
		tmp = t_0;
	} else if (re <= 8.7e-16) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 8.8e+138) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0028:
		tmp = t_0
	elif re <= 8.7e-16:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 8.8e+138:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0028)
		tmp = t_0;
	elseif (re <= 8.7e-16)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 8.8e+138)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0028)
		tmp = t_0;
	elseif (re <= 8.7e-16)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 8.8e+138)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0028], t$95$0, If[LessEqual[re, 8.7e-16], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.8e+138], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00279999999999999997 or 8.70000000000000036e-16 < re < 8.8000000000000003e138

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.3%

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

   if -0.00279999999999999997 < re < 8.70000000000000036e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-rgt1-in 99.1%

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

   4. Simplified 99.1%

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

   if 8.8000000000000003e138 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 97.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 97.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 97.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 97.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 97.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 97.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 97.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 97.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 97.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 97.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 97.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 97.0%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 97.0%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 97.0%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 97.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0028:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00375 \lor \neg \left(re \leq 8.7 \cdot 10^{-16}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00375) (not (<= re 8.7e-16)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00375) || !(re <= 8.7e-16)) {
		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 ((re <= (-0.00375d0)) .or. (.not. (re <= 8.7d-16))) 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 ((re <= -0.00375) || !(re <= 8.7e-16)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00375) or not (re <= 8.7e-16):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00375) || !(re <= 8.7e-16))
		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 ((re <= -0.00375) || ~((re <= 8.7e-16)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00375], N[Not[LessEqual[re, 8.7e-16]], $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 re < -0.0037499999999999999 or 8.70000000000000036e-16 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.8%

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

   if -0.0037499999999999999 < re < 8.70000000000000036e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-rgt1-in 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00375 \lor \neg \left(re \leq 8.7 \cdot 10^{-16}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 7: 65.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{+120}:\\ \;\;\;\;re \cdot \left(re \cdot \frac{-im}{1 - re}\right) - \frac{im}{re + -1}\\ \mathbf{elif}\;re \leq 3.1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.6e+120)
   (- (* re (* re (/ (- im) (- 1.0 re)))) (/ im (+ re -1.0)))
   (if (<= re 3.1) (sin im) (* im (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.6e+120) {
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0));
	} else if (re <= 3.1) {
		tmp = sin(im);
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9.6d+120)) then
        tmp = (re * (re * (-im / (1.0d0 - re)))) - (im / (re + (-1.0d0)))
    else if (re <= 3.1d0) then
        tmp = sin(im)
    else
        tmp = im * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.6e+120) {
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0));
	} else if (re <= 3.1) {
		tmp = Math.sin(im);
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.6e+120:
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0))
	elif re <= 3.1:
		tmp = math.sin(im)
	else:
		tmp = im * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.6e+120)
		tmp = Float64(Float64(re * Float64(re * Float64(Float64(-im) / Float64(1.0 - re)))) - Float64(im / Float64(re + -1.0)));
	elseif (re <= 3.1)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.6e+120)
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0));
	elseif (re <= 3.1)
		tmp = sin(im);
	else
		tmp = im * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.6e+120], N[(N[(re * N[(re * N[((-im) / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im / N[(re + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.1], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.60000000000000004e120

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   3. Taylor expanded in re around 0 2.5%

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

      1. distribute-lft1-in 2.5%

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

      2. +-commutative 2.5%

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

      3. *-commutative 2.5%

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

      4. flip-+ 1.9%

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

      5. associate-*r/ 1.9%

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

      6. metadata-eval 1.9%

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

   5. Applied egg-rr 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 9.9%

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

   7. Simplified 9.9%

      \[\leadsto \color{blue}{\frac{1 - re \cdot re}{\frac{1 - re}{im}}} \]
   8. Step-by-step derivation

      1. div-sub 9.9%

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

      2. frac-2neg 9.9%

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

      3. associate-/r/ 9.9%

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

      4. fma-neg 9.9%

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

      5. div-inv 9.9%

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

      6. clear-num 9.9%

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

   9. Applied egg-rr 9.9%

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

       1. fma-udef 9.9%

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

       2. +-commutative 9.9%

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

       3. distribute-rgt-neg-out 9.9%

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

       4. associate-/r/ 9.9%

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

       5. unsub-neg 9.9%

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

       6. distribute-rgt-neg-in 9.9%

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

       7. associate-*l* 55.3%

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

       8. distribute-neg-frac 55.3%

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

       9. associate-/r/ 55.3%

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

       10. associate-*l/ 55.3%

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

       11. *-lft-identity 55.3%

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

       12. sub-neg 55.3%

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

       13. +-commutative 55.3%

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

       14. distribute-neg-in 55.3%

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

       15. remove-double-neg 55.3%

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

       16. metadata-eval 55.3%

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

   11. Simplified 55.3%

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

   if -9.60000000000000004e120 < re < 3.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 76.6%

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

   if 3.10000000000000009 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.5%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 53.5%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 53.5%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 53.5%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 53.5%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 53.5%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 53.5%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 53.5%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 53.5%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 53.5%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 53.5%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 53.5%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 53.5%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 53.5%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 53.5%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 53.5%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 53.5%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 47.7%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.7%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} \]

      2. associate-*l* 47.7%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(im \cdot 0.5\right)} \]

      3. unpow2 47.7%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot 0.5\right) \]

   10. Simplified 47.7%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]

   11. Taylor expanded in re around 0 47.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   12. Step-by-step derivation

       1. unpow2 47.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) \]

       2. associate-*r* 47.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot im} \]

       3. *-commutative 47.7%

          \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]

   13. Simplified 47.7%

       \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{+120}:\\ \;\;\;\;re \cdot \left(re \cdot \frac{-im}{1 - re}\right) - \frac{im}{re + -1}\\ \mathbf{elif}\;re \leq 3.1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 42.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;re \cdot \left(re \cdot \frac{-im}{1 - re}\right) - \frac{im}{re + -1}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.84)
   (- (* re (* re (/ (- im) (- 1.0 re)))) (/ im (+ re -1.0)))
   (* im (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0));
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.84d0) then
        tmp = (re * (re * (-im / (1.0d0 - re)))) - (im / (re + (-1.0d0)))
    else
        tmp = im * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0));
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.84:
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0))
	else:
		tmp = im * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.84)
		tmp = Float64(Float64(re * Float64(re * Float64(Float64(-im) / Float64(1.0 - re)))) - Float64(im / Float64(re + -1.0)));
	else
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.84)
		tmp = (re * (re * (-im / (1.0 - re)))) - (im / (re + -1.0));
	else
		tmp = im * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.84], N[(N[(re * N[(re * N[((-im) / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im / N[(re + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.839999999999999969

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.0%

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

   3. Taylor expanded in re around 0 30.7%

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

      1. distribute-lft1-in 30.7%

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

      2. +-commutative 30.7%

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

      3. *-commutative 30.7%

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

      4. flip-+ 30.6%

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

      5. associate-*r/ 30.6%

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

      6. metadata-eval 30.6%

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

   5. Applied egg-rr 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-/l* 32.8%

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

   7. Simplified 32.8%

      \[\leadsto \color{blue}{\frac{1 - re \cdot re}{\frac{1 - re}{im}}} \]
   8. Step-by-step derivation

      1. div-sub 32.8%

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

      2. frac-2neg 32.8%

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

      3. associate-/r/ 32.8%

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

      4. fma-neg 32.8%

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

      5. div-inv 32.8%

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

      6. clear-num 32.4%

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

   9. Applied egg-rr 32.4%

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

       1. fma-udef 32.4%

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

       2. +-commutative 32.4%

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

       3. distribute-rgt-neg-out 32.4%

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

       4. associate-/r/ 32.3%

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

       5. unsub-neg 32.3%

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

       6. distribute-rgt-neg-in 32.3%

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

       7. associate-*l* 42.5%

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

       8. distribute-neg-frac 42.5%

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

       9. associate-/r/ 42.6%

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

       10. associate-*l/ 42.6%

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

       11. *-lft-identity 42.6%

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

       12. sub-neg 42.6%

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

       13. +-commutative 42.6%

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

       14. distribute-neg-in 42.6%

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

       15. remove-double-neg 42.6%

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

       16. metadata-eval 42.6%

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

   11. Simplified 42.6%

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

   if 0.839999999999999969 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 53.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 53.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 53.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 53.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 53.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 53.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 53.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 47.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.0%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} \]

      2. associate-*l* 47.0%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(im \cdot 0.5\right)} \]

      3. unpow2 47.0%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot 0.5\right) \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]

   11. Taylor expanded in re around 0 47.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   12. Step-by-step derivation

       1. unpow2 47.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) \]

       2. associate-*r* 47.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot im} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]

   13. Simplified 47.0%

       \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;re \cdot \left(re \cdot \frac{-im}{1 - re}\right) - \frac{im}{re + -1}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 38.0% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1}{im} - \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.84)
   (/ (- 1.0 (* re re)) (- (/ 1.0 im) (/ re im)))
   (* im (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = (1.0 - (re * re)) / ((1.0 / im) - (re / im));
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.84d0) then
        tmp = (1.0d0 - (re * re)) / ((1.0d0 / im) - (re / im))
    else
        tmp = im * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = (1.0 - (re * re)) / ((1.0 / im) - (re / im));
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.84:
		tmp = (1.0 - (re * re)) / ((1.0 / im) - (re / im))
	else:
		tmp = im * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.84)
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(Float64(1.0 / im) - Float64(re / im)));
	else
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.84)
		tmp = (1.0 - (re * re)) / ((1.0 / im) - (re / im));
	else
		tmp = im * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.84], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / im), $MachinePrecision] - N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.839999999999999969

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.0%

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

   3. Taylor expanded in re around 0 30.7%

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

      1. distribute-lft1-in 30.7%

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

      2. +-commutative 30.7%

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

      3. *-commutative 30.7%

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

      4. flip-+ 30.6%

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

      5. associate-*r/ 30.6%

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

      6. metadata-eval 30.6%

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

   5. Applied egg-rr 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-/l* 32.8%

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

   7. Simplified 32.8%

      \[\leadsto \color{blue}{\frac{1 - re \cdot re}{\frac{1 - re}{im}}} \]
   8. Step-by-step derivation

      1. div-sub 32.8%

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

   9. Applied egg-rr 32.8%

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

   if 0.839999999999999969 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 53.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 53.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 53.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 53.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 53.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 53.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 53.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 47.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.0%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} \]

      2. associate-*l* 47.0%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(im \cdot 0.5\right)} \]

      3. unpow2 47.0%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot 0.5\right) \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]

   11. Taylor expanded in re around 0 47.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   12. Step-by-step derivation

       1. unpow2 47.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) \]

       2. associate-*r* 47.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot im} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]

   13. Simplified 47.0%

       \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1}{im} - \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 38.0% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1 - re}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.84)
   (/ (- 1.0 (* re re)) (/ (- 1.0 re) im))
   (* im (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.84d0) then
        tmp = (1.0d0 - (re * re)) / ((1.0d0 - re) / im)
    else
        tmp = im * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.84:
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im)
	else:
		tmp = im * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.84)
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(Float64(1.0 - re) / im));
	else
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.84)
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	else
		tmp = im * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.84], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.839999999999999969

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.0%

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

   3. Taylor expanded in re around 0 30.7%

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

      1. distribute-lft1-in 30.7%

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

      2. +-commutative 30.7%

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

      3. *-commutative 30.7%

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

      4. flip-+ 30.6%

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

      5. associate-*r/ 30.6%

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

      6. metadata-eval 30.6%

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

   5. Applied egg-rr 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-/l* 32.8%

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

   7. Simplified 32.8%

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

   if 0.839999999999999969 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 53.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 53.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 53.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 53.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 53.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 53.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 53.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 47.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.0%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} \]

      2. associate-*l* 47.0%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(im \cdot 0.5\right)} \]

      3. unpow2 47.0%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot 0.5\right) \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]

   11. Taylor expanded in re around 0 47.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   12. Step-by-step derivation

       1. unpow2 47.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) \]

       2. associate-*r* 47.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot im} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]

   13. Simplified 47.0%

       \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1 - re}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 36.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.84) im (* im (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = im;
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.84d0) then
        tmp = im
    else
        tmp = im * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		tmp = im;
	} else {
		tmp = im * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.84:
		tmp = im
	else:
		tmp = im * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.84)
		tmp = im;
	else
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.84)
		tmp = im;
	else
		tmp = im * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.84], im, N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.839999999999999969

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.0%

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

   3. Taylor expanded in re around 0 30.8%

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

   if 0.839999999999999969 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 53.0%

         \[\leadsto \color{blue}{\left(\sin im + \sin im \cdot re\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]

      2. +-commutative 53.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot re + \sin im\right)} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      3. *-commutative 53.0%

         \[\leadsto \left(\color{blue}{re \cdot \sin im} + \sin im\right) + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      4. distribute-lft1-in 53.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right) \]

      5. *-commutative 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \sin im\right)} \]

      6. associate-*r* 53.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      7. distribute-rgt-out 53.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{{re}^{2} \cdot 0.5}\right) \]

      9. unpow2 53.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) \]

      10. associate-*l* 53.0%

          \[\leadsto \sin im \cdot \left(\left(re + 1\right) + \color{blue}{re \cdot \left(re \cdot 0.5\right)}\right) \]

   4. Simplified 53.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)} \]

   5. Taylor expanded in re around inf 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \sin im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      2. *-commutative 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      3. associate-*r* 52.9%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified 52.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 47.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.0%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} \]

      2. associate-*l* 47.0%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(im \cdot 0.5\right)} \]

      3. unpow2 47.0%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot 0.5\right) \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]

   11. Taylor expanded in re around 0 47.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   12. Step-by-step derivation

       1. unpow2 47.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) \]

       2. associate-*r* 47.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot im} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]

   13. Simplified 47.0%

       \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 29.3% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 0.84) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.84) {
		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 <= 0.84d0) 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 <= 0.84) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.84:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.84)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.84)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.84], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.839999999999999969

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.0%

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

   3. Taylor expanded in re around 0 30.8%

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

   if 0.839999999999999969 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.4%

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

   3. Taylor expanded in re around 0 15.2%

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

   4. Taylor expanded in re around inf 15.2%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.84:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 13: 29.2% 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. Taylor expanded in re around 0 47.9%

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

   1. *-commutative 47.9%

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

   2. distribute-rgt1-in 47.9%

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

4. Simplified 47.9%

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

5. Taylor expanded in im around 0 27.1%

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

6. Final simplification 27.1%

   \[\leadsto im \cdot \left(re + 1\right) \]

Alternative 14: 29.2% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 71.2%

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

3. Taylor expanded in re around 0 27.1%

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

4. Final simplification 27.1%

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

Alternative 15: 26.4% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 71.2%

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

3. Taylor expanded in re around 0 24.4%

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

4. Final simplification 24.4%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023201 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))