math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 15

Speedup: 1.0×

• 25.2% 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: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999998 \lor \neg \left(e^{re} \leq 1.00000002\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999999998) (not (<= (exp re) 1.00000002)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999999998) || !(exp(re) <= 1.00000002)) {
		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) <= 0.999999998d0) .or. (.not. (exp(re) <= 1.00000002d0))) 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) <= 0.999999998) || !(Math.exp(re) <= 1.00000002)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.999999998) or not (math.exp(re) <= 1.00000002):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.999999998) || !(exp(re) <= 1.00000002))
		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) <= 0.999999998) || ~((exp(re) <= 1.00000002)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.999999998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.00000002]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999997999999946 or 1.0000000200000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.2%

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

   if 0.999999997999999946 < (exp.f64 re) < 1.0000000200000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999998 \lor \neg \left(e^{re} \leq 1.00000002\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.059 \lor \neg \left(re \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6)
   0.0
   (if (or (<= re 0.059) (not (<= re 1.02e+103)))
     (*
      (sin im)
      (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = 0.0;
	} else if ((re <= 0.059) || !(re <= 1.02e+103)) {
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.6d0)) then
        tmp = 0.0d0
    else if ((re <= 0.059d0) .or. (.not. (re <= 1.02d+103))) then
        tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = 0.0;
	} else if ((re <= 0.059) || !(re <= 1.02e+103)) {
		tmp = Math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6:
		tmp = 0.0
	elif (re <= 0.059) or not (re <= 1.02e+103):
		tmp = math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6)
		tmp = 0.0;
	elseif ((re <= 0.059) || !(re <= 1.02e+103))
		tmp = Float64(sin(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6)
		tmp = 0.0;
	elseif ((re <= 0.059) || ~((re <= 1.02e+103)))
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], 0.0, If[Or[LessEqual[re, 0.059], N[Not[LessEqual[re, 1.02e+103]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1.6000000000000001 < re < 0.058999999999999997 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. distribute-rgt1-in 99.8%

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

      5. associate-*r* 99.8%

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

      6. associate-*r* 99.8%

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

      7. distribute-rgt-out 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   if 0.058999999999999997 < re < 1.01999999999999991e103

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 79.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.059 \lor \neg \left(re \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -80:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -80.0)
     0.0
     (if (<= re 4.4e-8)
       (* (sin im) (+ (+ re 1.0) t_0))
       (if (<= re 1.9e+154) (* (exp re) im) (* (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -80.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re) * im;
	} 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) :: tmp
    t_0 = re * (re * 0.5d0)
    if (re <= (-80.0d0)) then
        tmp = 0.0d0
    else if (re <= 4.4d-8) then
        tmp = sin(im) * ((re + 1.0d0) + t_0)
    else if (re <= 1.9d+154) then
        tmp = exp(re) * im
    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 tmp;
	if (re <= -80.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = Math.sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -80.0:
		tmp = 0.0
	elif re <= 4.4e-8:
		tmp = math.sin(im) * ((re + 1.0) + t_0)
	elif re <= 1.9e+154:
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -80.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 1.9e+154)
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -80.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = sin(im) * ((re + 1.0) + t_0);
	elseif (re <= 1.9e+154)
		tmp = exp(re) * im;
	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]}, If[LessEqual[re, -80.0], 0.0, If[LessEqual[re, 4.4e-8], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -80

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -80 < re < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. distribute-rgt1-in 99.8%

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

      5. associate-*r* 99.8%

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

      6. associate-*r* 99.8%

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

      7. distribute-rgt-out 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in re around 0 99.7%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   7. Simplified 99.7%

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

   if 4.3999999999999997e-8 < re < 1.8999999999999999e154

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 84.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. associate-*r* 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 100.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -80:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 4.4e-8)
     (* (sin im) (+ re 1.0))
     (if (<= re 1.9e+154) (* (exp re) im) (* (sin im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re) * im;
	} 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) :: tmp
    if (re <= (-1.0d0)) then
        tmp = 0.0d0
    else if (re <= 4.4d-8) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 4.4e-8:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 4.4e-8], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.6%

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

   4. Simplified 99.6%

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

   if 4.3999999999999997e-8 < re < 1.8999999999999999e154

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 84.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. associate-*r* 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 100.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 4.4e-8) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.0d0)) then
        tmp = 0.0d0
    else if (re <= 4.4d-8) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 4.4e-8:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 4.4e-8], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.6%

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

   4. Simplified 99.6%

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

   if 4.3999999999999997e-8 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 81.4%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 7: 84.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -115:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -115.0)
   0.0
   (if (<= re 4.4e-8) (sin im) (+ im (* im (* re (+ 1.0 (* re 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -115.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (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 <= (-115.0d0)) then
        tmp = 0.0d0
    else if (re <= 4.4d-8) then
        tmp = sin(im)
    else
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -115.0) {
		tmp = 0.0;
	} else if (re <= 4.4e-8) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -115.0:
		tmp = 0.0
	elif re <= 4.4e-8:
		tmp = math.sin(im)
	else:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -115.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -115.0)
		tmp = 0.0;
	elseif (re <= 4.4e-8)
		tmp = sin(im);
	else
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -115.0], 0.0, If[LessEqual[re, 4.4e-8], N[Sin[im], $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -115

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -115 < re < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

   if 4.3999999999999997e-8 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 68.7%

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

      1. associate-+r+ 68.7%

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

      2. +-commutative 68.7%

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

      3. associate-+r+ 68.7%

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

      4. distribute-rgt1-in 68.7%

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

      5. associate-*r* 68.7%

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

      6. associate-*r* 68.7%

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

      7. distribute-rgt-out 68.7%

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

      8. *-commutative 68.7%

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

      9. distribute-rgt-out 68.7%

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

      10. +-commutative 68.7%

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

   4. Simplified 67.3%

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

   5. Taylor expanded in re around 0 48.0%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 48.0%

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

      2. +-commutative 48.0%

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

      3. unpow2 48.0%

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

      4. *-commutative 48.0%

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

      5. associate-*r* 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in im around 0 40.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 40.4%

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

      2. *-rgt-identity 40.4%

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

      3. +-commutative 40.4%

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

      4. unpow2 40.4%

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

      5. fma-def 40.4%

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

   10. Simplified 40.4%

       \[\leadsto \color{blue}{im + im \cdot \mathsf{fma}\left(0.5, re \cdot re, re\right)} \]
   11. Step-by-step derivation

       1. fma-udef 40.4%

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

       2. *-commutative 40.4%

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

       3. associate-*r* 40.4%

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

       4. +-commutative 40.4%

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

       5. *-commutative 40.4%

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

       6. distribute-rgt1-in 40.4%

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

       7. *-commutative 40.4%

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

   12. Applied egg-rr 40.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -115:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 61.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im + re \cdot im\\ t_1 := im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \mathbf{if}\;re \leq -125:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+114}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+147}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;im + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ im (* re im))) (t_1 (* im (* (* re re) 0.5))))
   (if (<= re -125.0)
     0.0
     (if (<= re 1.95e+114)
       (+ im (* im (* re (+ 1.0 (* re 0.5)))))
       (if (<= re 1.15e+147)
         (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))
         (+ im (* (* re re) (* im 0.5))))))))

C

double code(double re, double im) {
	double t_0 = im + (re * im);
	double t_1 = im * ((re * re) * 0.5);
	double tmp;
	if (re <= -125.0) {
		tmp = 0.0;
	} else if (re <= 1.95e+114) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.15e+147) {
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
	} else {
		tmp = im + ((re * re) * (im * 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) :: t_1
    real(8) :: tmp
    t_0 = im + (re * im)
    t_1 = im * ((re * re) * 0.5d0)
    if (re <= (-125.0d0)) then
        tmp = 0.0d0
    else if (re <= 1.95d+114) then
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1.15d+147) then
        tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
    else
        tmp = im + ((re * re) * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im + (re * im);
	double t_1 = im * ((re * re) * 0.5);
	double tmp;
	if (re <= -125.0) {
		tmp = 0.0;
	} else if (re <= 1.95e+114) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.15e+147) {
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
	} else {
		tmp = im + ((re * re) * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im + (re * im)
	t_1 = im * ((re * re) * 0.5)
	tmp = 0
	if re <= -125.0:
		tmp = 0.0
	elif re <= 1.95e+114:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	elif re <= 1.15e+147:
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
	else:
		tmp = im + ((re * re) * (im * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im + Float64(re * im))
	t_1 = Float64(im * Float64(Float64(re * re) * 0.5))
	tmp = 0.0
	if (re <= -125.0)
		tmp = 0.0;
	elseif (re <= 1.95e+114)
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1.15e+147)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1));
	else
		tmp = Float64(im + Float64(Float64(re * re) * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im + (re * im);
	t_1 = im * ((re * re) * 0.5);
	tmp = 0.0;
	if (re <= -125.0)
		tmp = 0.0;
	elseif (re <= 1.95e+114)
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	elseif (re <= 1.15e+147)
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
	else
		tmp = im + ((re * re) * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -125.0], 0.0, If[LessEqual[re, 1.95e+114], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+147], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(im + N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -125

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -125 < re < 1.95e114

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 86.9%

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

      1. associate-+r+ 86.9%

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

      2. +-commutative 86.9%

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

      3. associate-+r+ 86.9%

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

      4. distribute-rgt1-in 86.9%

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

      5. associate-*r* 86.9%

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

      6. associate-*r* 86.9%

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

      7. distribute-rgt-out 86.9%

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

      8. *-commutative 86.9%

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

      9. distribute-rgt-out 86.9%

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

      10. +-commutative 86.9%

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

   4. Simplified 86.3%

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

   5. Taylor expanded in re around 0 83.8%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 83.8%

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

      2. +-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. *-commutative 83.8%

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

      5. associate-*r* 83.8%

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

   7. Simplified 83.8%

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

   8. Taylor expanded in im around 0 42.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 42.4%

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

      2. *-rgt-identity 42.4%

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

      3. +-commutative 42.4%

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

      4. unpow2 42.4%

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

      5. fma-def 42.4%

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

   10. Simplified 42.4%

       \[\leadsto \color{blue}{im + im \cdot \mathsf{fma}\left(0.5, re \cdot re, re\right)} \]
   11. Step-by-step derivation

       1. fma-udef 42.4%

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

       2. *-commutative 42.4%

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

       3. associate-*r* 42.4%

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

       4. +-commutative 42.4%

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

       5. *-commutative 42.4%

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

       6. distribute-rgt1-in 42.4%

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

       7. *-commutative 42.4%

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

   12. Applied egg-rr 42.4%

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

   if 1.95e114 < re < 1.15e147

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. associate-*r* 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 5.3%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 5.3%

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

      2. +-commutative 5.3%

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

      3. unpow2 5.3%

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

      4. *-commutative 5.3%

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

      5. associate-*r* 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in im around 0 4.5%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 4.5%

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

      2. *-rgt-identity 4.5%

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

      3. +-commutative 4.5%

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

      4. unpow2 4.5%

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

      5. fma-def 4.5%

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

   10. Simplified 4.5%

       \[\leadsto \color{blue}{im + im \cdot \mathsf{fma}\left(0.5, re \cdot re, re\right)} \]
   11. Step-by-step derivation

       1. *-commutative 4.5%

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

       2. distribute-rgt1-in 4.5%

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

       3. fma-udef 4.5%

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

       4. *-commutative 4.5%

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

       5. associate-*r* 4.5%

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

       6. associate-+r+ 4.5%

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

       7. +-commutative 4.5%

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

       8. +-commutative 4.5%

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

       9. *-commutative 4.5%

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

       10. distribute-lft-in 4.5%

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

       11. flip-+ 56.7%

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

   12. Applied egg-rr 56.7%

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

   if 1.15e147 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. associate-*r* 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 96.9%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 96.9%

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

      2. +-commutative 96.9%

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

      3. unpow2 96.9%

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

      4. *-commutative 96.9%

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

      5. associate-*r* 96.9%

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

   7. Simplified 96.9%

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

   8. Taylor expanded in im around 0 74.3%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 74.3%

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

      2. *-rgt-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-def 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in re around inf 74.3%

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

       1. *-commutative 74.3%

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

       2. unpow2 74.3%

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

       3. *-commutative 74.3%

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

       4. associate-*l* 74.3%

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

   13. Simplified 74.3%

       \[\leadsto im + \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -125:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+114}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(im + re \cdot im\right) \cdot \left(im + re \cdot im\right) - \left(im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\right)}{\left(im + re \cdot im\right) - im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;im + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 61.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -105:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -105.0) 0.0 (* im (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -105.0) {
		tmp = 0.0;
	} else {
		tmp = im * ((re + 1.0) + (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 <= (-105.0d0)) then
        tmp = 0.0d0
    else
        tmp = im * ((re + 1.0d0) + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -105.0) {
		tmp = 0.0;
	} else {
		tmp = im * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -105.0:
		tmp = 0.0
	else:
		tmp = im * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -105.0)
		tmp = 0.0;
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -105.0)
		tmp = 0.0;
	else
		tmp = im * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -105.0], 0.0, N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -105

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -105 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 89.4%

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

      1. associate-+r+ 89.4%

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

      2. +-commutative 89.4%

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

      3. associate-+r+ 89.4%

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

      4. distribute-rgt1-in 89.4%

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

      5. associate-*r* 89.4%

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

      6. associate-*r* 89.4%

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

      7. distribute-rgt-out 89.4%

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

      8. *-commutative 89.4%

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

      9. distribute-rgt-out 89.4%

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

      10. +-commutative 89.4%

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

   4. Simplified 88.9%

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

   5. Taylor expanded in re around 0 82.4%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 82.4%

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

      2. +-commutative 82.4%

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

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

   7. Simplified 82.4%

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

   8. Taylor expanded in im around 0 45.5%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 45.5%

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

      2. unpow2 45.5%

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

      3. *-commutative 45.5%

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

      4. associate-*r* 45.5%

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

   10. Simplified 45.5%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -105:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 61.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -65:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -65.0) 0.0 (+ im (* im (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -65.0) {
		tmp = 0.0;
	} else {
		tmp = im + (im * (re * (1.0 + (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 <= (-65.0d0)) then
        tmp = 0.0d0
    else
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -65.0) {
		tmp = 0.0;
	} else {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -65.0:
		tmp = 0.0
	else:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -65.0)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -65.0)
		tmp = 0.0;
	else
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -65.0], 0.0, N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -65

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -65 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 89.4%

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

      1. associate-+r+ 89.4%

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

      2. +-commutative 89.4%

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

      3. associate-+r+ 89.4%

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

      4. distribute-rgt1-in 89.4%

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

      5. associate-*r* 89.4%

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

      6. associate-*r* 89.4%

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

      7. distribute-rgt-out 89.4%

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

      8. *-commutative 89.4%

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

      9. distribute-rgt-out 89.4%

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

      10. +-commutative 89.4%

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

   4. Simplified 88.9%

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

   5. Taylor expanded in re around 0 82.4%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 82.4%

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

      2. +-commutative 82.4%

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

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

   7. Simplified 82.4%

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

   8. Taylor expanded in im around 0 45.5%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 45.5%

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

      2. *-rgt-identity 45.5%

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

      3. +-commutative 45.5%

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

      4. unpow2 45.5%

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

      5. fma-def 45.5%

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

   10. Simplified 45.5%

       \[\leadsto \color{blue}{im + im \cdot \mathsf{fma}\left(0.5, re \cdot re, re\right)} \]
   11. Step-by-step derivation

       1. fma-udef 45.5%

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

       2. *-commutative 45.5%

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

       3. associate-*r* 45.5%

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

       4. +-commutative 45.5%

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

       5. *-commutative 45.5%

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

       6. distribute-rgt1-in 45.5%

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

       7. *-commutative 45.5%

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

   12. Applied egg-rr 45.5%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -65:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 11: 61.0% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -98:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -98.0) 0.0 (+ im (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -98.0) {
		tmp = 0.0;
	} else {
		tmp = im + ((re * re) * (im * 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 <= (-98.0d0)) then
        tmp = 0.0d0
    else
        tmp = im + ((re * re) * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -98.0) {
		tmp = 0.0;
	} else {
		tmp = im + ((re * re) * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -98.0:
		tmp = 0.0
	else:
		tmp = im + ((re * re) * (im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -98.0)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(Float64(re * re) * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -98.0)
		tmp = 0.0;
	else
		tmp = im + ((re * re) * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -98.0], 0.0, N[(im + N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -98

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -98 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 89.4%

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

      1. associate-+r+ 89.4%

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

      2. +-commutative 89.4%

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

      3. associate-+r+ 89.4%

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

      4. distribute-rgt1-in 89.4%

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

      5. associate-*r* 89.4%

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

      6. associate-*r* 89.4%

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

      7. distribute-rgt-out 89.4%

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

      8. *-commutative 89.4%

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

      9. distribute-rgt-out 89.4%

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

      10. +-commutative 89.4%

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

   4. Simplified 88.9%

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

   5. Taylor expanded in re around 0 82.4%

      \[\leadsto \sin im \cdot \color{blue}{\left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 82.4%

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

      2. +-commutative 82.4%

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

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

   7. Simplified 82.4%

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

   8. Taylor expanded in im around 0 45.5%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 45.5%

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

      2. *-rgt-identity 45.5%

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

      3. +-commutative 45.5%

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

      4. unpow2 45.5%

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

      5. fma-def 45.5%

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

   10. Simplified 45.5%

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

   11. Taylor expanded in re around inf 44.9%

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

       1. *-commutative 44.9%

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

       2. unpow2 44.9%

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

       3. *-commutative 44.9%

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

       4. associate-*l* 44.9%

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

   13. Simplified 44.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -98:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 53.5% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -58:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -58.0) 0.0 (if (<= re 1.0) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -58.0) {
		tmp = 0.0;
	} else if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-58.0d0)) then
        tmp = 0.0d0
    else if (re <= 1.0d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -58.0) {
		tmp = 0.0;
	} else if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -58.0:
		tmp = 0.0
	elif re <= 1.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -58.0)
		tmp = 0.0;
	elseif (re <= 1.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -58.0)
		tmp = 0.0;
	elseif (re <= 1.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -58.0], 0.0, If[LessEqual[re, 1.0], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -58

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -58 < re < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 48.8%

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

   3. Taylor expanded in re around 0 47.8%

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

   if 1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.4%

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

      1. distribute-rgt1-in 4.4%

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

   4. Simplified 4.4%

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

   5. Taylor expanded in re around inf 4.4%

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

      1. *-commutative 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in im around 0 10.3%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -58:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 13: 53.8% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -1.3) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.3) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.3d0)) then
        tmp = 0.0d0
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.3) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.3:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.3)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.3)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.3], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.30000000000000004

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1.30000000000000004 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 59.2%

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

   3. Taylor expanded in re around 0 36.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 14: 29.5% 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 100.0%

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

   2. Taylor expanded in im around 0 61.6%

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

   3. Taylor expanded in re around 0 36.7%

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

   if 1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.4%

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

      1. distribute-rgt1-in 4.4%

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

   4. Simplified 4.4%

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

   5. Taylor expanded in re around inf 4.4%

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

      1. *-commutative 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in im around 0 10.3%

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

4. Final simplification 29.7%

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

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 66.7%

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

3. Taylor expanded in re around 0 27.6%

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

4. Final simplification 27.6%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023275 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))