math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 14

Speedup: 1.0×

• 24.4% 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.0000000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.0000000001)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.0000000001)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 1.0000000001d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 1.0000000001)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.0000000001):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.0000000001))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 1.0000000001)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0000000001]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.0000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 70.4%

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

   if 1 < (exp.f64 re) < 1.0000000001

   1. Initial program 99.0%

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

   2. Taylor expanded in re around 0 83.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.0000000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0051:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0051)
     t_0
     (if (<= re 1.35e-10)
       (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
       (if (<= re 1.45e+100)
         t_0
         (*
          (sin im)
          (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0051) {
		tmp = t_0;
	} else if (re <= 1.35e-10) {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.45e+100) {
		tmp = t_0;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	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 <= (-0.0051d0)) then
        tmp = t_0
    else if (re <= 1.35d-10) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 1.45d+100) then
        tmp = t_0
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    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 <= -0.0051) {
		tmp = t_0;
	} else if (re <= 1.35e-10) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.45e+100) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0051:
		tmp = t_0
	elif re <= 1.35e-10:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 1.45e+100:
		tmp = t_0
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0051)
		tmp = t_0;
	elseif (re <= 1.35e-10)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 1.45e+100)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0051)
		tmp = t_0;
	elseif (re <= 1.35e-10)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 1.45e+100)
		tmp = t_0;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0051], t$95$0, If[LessEqual[re, 1.35e-10], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.45e+100], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0051000000000000004 or 1.35e-10 < re < 1.45e100

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.3%

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

   if -0.0051000000000000004 < re < 1.35e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-out 99.9%

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

      8. *-commutative 99.9%

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

      9. unpow2 99.9%

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

      10. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

   if 1.45e100 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.3%

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

      1. associate-+r+ 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-rgt1-in 97.3%

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

      4. *-commutative 97.3%

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

      5. +-commutative 97.3%

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

      6. *-commutative 97.3%

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

      7. associate-*r* 97.3%

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

      8. *-commutative 97.3%

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

      9. associate-*r* 97.3%

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

      10. distribute-rgt-out 97.3%

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

      11. distribute-lft-out 97.3%

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

      12. +-commutative 97.3%

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

   4. Simplified 97.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0051:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+100}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0122:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (* (exp re) im)))
   (if (<= re -0.0122)
     t_1
     (if (<= re 1.35e-10)
       (* (sin im) (+ (+ re 1.0) t_0))
       (if (<= re 1.9e+154) 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 <= -0.0122) {
		tmp = t_1;
	} else if (re <= 1.35e-10) {
		tmp = sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		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 <= (-0.0122d0)) then
        tmp = t_1
    else if (re <= 1.35d-10) then
        tmp = sin(im) * ((re + 1.0d0) + t_0)
    else if (re <= 1.9d+154) 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 <= -0.0122) {
		tmp = t_1;
	} else if (re <= 1.35e-10) {
		tmp = Math.sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		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 <= -0.0122:
		tmp = t_1
	elif re <= 1.35e-10:
		tmp = math.sin(im) * ((re + 1.0) + t_0)
	elif re <= 1.9e+154:
		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 <= -0.0122)
		tmp = t_1;
	elseif (re <= 1.35e-10)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 1.9e+154)
		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 <= -0.0122)
		tmp = t_1;
	elseif (re <= 1.35e-10)
		tmp = sin(im) * ((re + 1.0) + t_0);
	elseif (re <= 1.9e+154)
		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, -0.0122], t$95$1, If[LessEqual[re, 1.35e-10], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$1, N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0122000000000000008 or 1.35e-10 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.9%

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

   if -0.0122000000000000008 < re < 1.35e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-out 99.9%

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

      8. *-commutative 99.9%

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

      9. unpow2 99.9%

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

      10. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0122:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -5.7e-5)
     t_0
     (if (<= re 1.35e-10)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.9e+154) 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 <= -5.7e-5) {
		tmp = t_0;
	} else if (re <= 1.35e-10) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		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 <= (-5.7d-5)) then
        tmp = t_0
    else if (re <= 1.35d-10) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) 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 <= -5.7e-5) {
		tmp = t_0;
	} else if (re <= 1.35e-10) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		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 <= -5.7e-5:
		tmp = t_0
	elif re <= 1.35e-10:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		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 <= -5.7e-5)
		tmp = t_0;
	elseif (re <= 1.35e-10)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		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 <= -5.7e-5)
		tmp = t_0;
	elseif (re <= 1.35e-10)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		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, -5.7e-5], t$95$0, If[LessEqual[re, 1.35e-10], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], 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 < -5.7000000000000003e-5 or 1.35e-10 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.9%

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

   if -5.7000000000000003e-5 < re < 1.35e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.8%

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

   4. Simplified 99.8%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00013 \lor \neg \left(re \leq 1.35 \cdot 10^{-10}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00013) (not (<= re 1.35e-10)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00013) || !(re <= 1.35e-10)) {
		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 <= (-0.00013d0)) .or. (.not. (re <= 1.35d-10))) 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 <= -0.00013) || !(re <= 1.35e-10)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00013) or not (re <= 1.35e-10):
		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 <= -0.00013) || !(re <= 1.35e-10))
		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 <= -0.00013) || ~((re <= 1.35e-10)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00013], N[Not[LessEqual[re, 1.35e-10]], $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 < -1.29999999999999989e-4 or 1.35e-10 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 83.9%

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

   if -1.29999999999999989e-4 < re < 1.35e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00013 \lor \neg \left(re \leq 1.35 \cdot 10^{-10}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 7: 61.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-10) (sin im) (/ (* im (- 1.0 (* re re))) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-10) {
		tmp = sin(im);
	} else {
		tmp = (im * (1.0 - (re * re))) / (1.0 - re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1d-10) then
        tmp = sin(im)
    else
        tmp = (im * (1.0d0 - (re * re))) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e-10) {
		tmp = Math.sin(im);
	} else {
		tmp = (im * (1.0 - (re * re))) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e-10:
		tmp = math.sin(im)
	else:
		tmp = (im * (1.0 - (re * re))) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-10)
		tmp = sin(im);
	else
		tmp = Float64(Float64(im * Float64(1.0 - Float64(re * re))) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e-10)
		tmp = sin(im);
	else
		tmp = (im * (1.0 - (re * re))) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-10], N[Sin[im], $MachinePrecision], N[(N[(im * N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.00000000000000004e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 62.3%

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

   if 1.00000000000000004e-10 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 6.4%

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

      1. *-commutative 6.4%

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

      2. distribute-rgt1-in 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in im around 0 14.4%

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

      1. flip-+ 28.6%

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

      2. associate-*l/ 31.6%

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

      3. metadata-eval 31.6%

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

   7. Applied egg-rr 31.6%

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

4. Final simplification 55.0%

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

Alternative 8: 36.4% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{+113}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(re \cdot \left(-re\right)\right)}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e+113) (+ im (* re im)) (/ (* im (* re (- re))) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e+113) {
		tmp = im + (re * im);
	} else {
		tmp = (im * (re * -re)) / (1.0 - re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2d+113) then
        tmp = im + (re * im)
    else
        tmp = (im * (re * -re)) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2e+113) {
		tmp = im + (re * im);
	} else {
		tmp = (im * (re * -re)) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2e+113:
		tmp = im + (re * im)
	else:
		tmp = (im * (re * -re)) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e+113)
		tmp = Float64(im + Float64(re * im));
	else
		tmp = Float64(Float64(im * Float64(re * Float64(-re))) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2e+113)
		tmp = im + (re * im);
	else
		tmp = (im * (re * -re)) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e+113], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(re * (-re)), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2e113

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 70.4%

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

   3. Taylor expanded in re around 0 30.9%

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

   if 2e113 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.2%

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

      1. *-commutative 5.2%

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

      2. distribute-rgt1-in 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in im around 0 20.8%

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

      1. flip-+ 52.7%

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

      2. associate-*l/ 59.6%

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

      3. metadata-eval 59.6%

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

   7. Applied egg-rr 59.6%

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

   8. Taylor expanded in re around inf 59.6%

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

      1. mul-1-neg 59.6%

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

      2. unpow2 59.6%

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

      3. *-commutative 59.6%

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

      4. distribute-rgt-neg-in 59.6%

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

   10. Simplified 59.6%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{+113}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(re \cdot \left(-re\right)\right)}{1 - re}\\ \end{array} \]

Alternative 9: 36.0% accurate, 18.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return im / ((re + -1.0) / ((re * 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)) / ((re * re) + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in re around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. distribute-rgt1-in 49.3%

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

4. Simplified 49.3%

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

5. Taylor expanded in im around 0 29.8%

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

   1. flip-+ 33.1%

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

   2. associate-*l/ 33.8%

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

   3. metadata-eval 33.8%

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

7. Applied egg-rr 33.8%

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

   1. frac-2neg 33.8%

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

   2. div-inv 33.8%

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

   3. *-commutative 33.8%

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

   4. distribute-rgt-neg-in 33.8%

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

9. Applied egg-rr 33.8%

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

    1. associate-*r/ 33.8%

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

    2. *-rgt-identity 33.8%

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

    3. associate-/l* 33.1%

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

    4. neg-sub0 33.1%

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

    5. associate--r- 33.1%

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

    6. metadata-eval 33.1%

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

    7. neg-sub0 33.1%

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

    8. associate--r- 33.1%

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

    9. metadata-eval 33.1%

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

11. Simplified 33.1%

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

12. Final simplification 33.1%

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

Alternative 10: 36.7% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in re around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. distribute-rgt1-in 49.3%

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

4. Simplified 49.3%

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

5. Taylor expanded in im around 0 29.8%

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

   1. flip-+ 33.1%

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

   2. associate-*l/ 33.8%

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

   3. metadata-eval 33.8%

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

7. Applied egg-rr 33.8%

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

8. Final simplification 33.8%

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

Alternative 11: 29.7% accurate, 40.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 71.1%

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

   3. Taylor expanded in re around 0 34.4%

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

   if 1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.1%

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

      1. *-commutative 4.1%

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

      2. distribute-rgt1-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 12.4%

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

   6. Taylor expanded in re around inf 12.4%

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

4. Final simplification 29.4%

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

Alternative 12: 29.6% 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 99.9%

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

2. Taylor expanded in re around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. distribute-rgt1-in 49.3%

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

4. Simplified 49.3%

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

5. Taylor expanded in im around 0 29.8%

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

6. Final simplification 29.8%

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

Alternative 13: 29.6% 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 99.9%

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

2. Taylor expanded in im around 0 70.0%

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

3. Taylor expanded in re around 0 29.8%

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

4. Final simplification 29.8%

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

Alternative 14: 26.5% 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 99.9%

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

2. Taylor expanded in im around 0 70.0%

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

3. Taylor expanded in re around 0 27.0%

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

4. Final simplification 27.0%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))