math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 13

Speedup: 1.0×

• 23.9% 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: 70.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.000001)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.000001)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 1.000001d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 1.000001)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.000001):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.000001))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 1.000001)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.000001]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.00000099999999992 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.4%

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

   if 1 < (exp.f64 re) < 1.00000099999999992

   1. Initial program 99.2%

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

   2. Taylor expanded in re around 0 77.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.027:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.027)
     t_0
     (if (<= re 1.05e-6)
       (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
       (if (<= re 6.2e+100)
         t_0
         (*
          (sin im)
          (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.027) {
		tmp = t_0;
	} else if (re <= 1.05e-6) {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 6.2e+100) {
		tmp = t_0;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.027d0)) then
        tmp = t_0
    else if (re <= 1.05d-6) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 6.2d+100) then
        tmp = t_0
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.027) {
		tmp = t_0;
	} else if (re <= 1.05e-6) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 6.2e+100) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.027:
		tmp = t_0
	elif re <= 1.05e-6:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 6.2e+100:
		tmp = t_0
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.027)
		tmp = t_0;
	elseif (re <= 1.05e-6)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 6.2e+100)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.027)
		tmp = t_0;
	elseif (re <= 1.05e-6)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 6.2e+100)
		tmp = t_0;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.027], t$95$0, If[LessEqual[re, 1.05e-6], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.2e+100], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0269999999999999997 or 1.0499999999999999e-6 < re < 6.20000000000000014e100

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.7%

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

   if -0.0269999999999999997 < re < 1.0499999999999999e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   if 6.20000000000000014e100 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.0%

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

      1. associate-+r+ 98.0%

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

      2. *-commutative 98.0%

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

      3. distribute-rgt1-in 98.0%

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

      4. *-commutative 98.0%

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

      5. +-commutative 98.0%

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

      6. *-commutative 98.0%

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

      7. associate-*r* 98.0%

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

      8. *-commutative 98.0%

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

      9. associate-*r* 98.0%

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

      10. distribute-rgt-out 98.0%

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

      11. distribute-lft-out 98.0%

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

      12. +-commutative 98.0%

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

   4. Simplified 98.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.027:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+100}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0022 \lor \neg \left(re \leq 1.05 \cdot 10^{-6}\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0022) (and (not (<= re 1.05e-6)) (<= re 1.9e+154)))
   (* (exp re) im)
   (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0022) || (!(re <= 1.05e-6) && (re <= 1.9e+154))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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 <= (-0.0022d0)) .or. (.not. (re <= 1.05d-6)) .and. (re <= 1.9d+154)) then
        tmp = exp(re) * im
    else
        tmp = sin(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 <= -0.0022) || (!(re <= 1.05e-6) && (re <= 1.9e+154))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0022) or (not (re <= 1.05e-6) and (re <= 1.9e+154)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0022) || (!(re <= 1.05e-6) && (re <= 1.9e+154)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(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 <= -0.0022) || (~((re <= 1.05e-6)) && (re <= 1.9e+154)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0022], And[N[Not[LessEqual[re, 1.05e-6]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * 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 < -0.00220000000000000013 or 1.0499999999999999e-6 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.1%

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

   if -0.00220000000000000013 < re < 1.0499999999999999e-6 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0022 \lor \neg \left(re \leq 1.05 \cdot 10^{-6}\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(re \cdot \sin im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -3.2e-8)
     t_0
     (if (<= re 1.05e-6)
       (* (sin im) (+ re 1.0))
       (if (<= re 5.2e+157) t_0 (* re (* 0.5 (* re (sin im)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -3.2e-8) {
		tmp = t_0;
	} else if (re <= 1.05e-6) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 5.2e+157) {
		tmp = t_0;
	} else {
		tmp = re * (0.5 * (re * sin(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-3.2d-8)) then
        tmp = t_0
    else if (re <= 1.05d-6) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 5.2d+157) then
        tmp = t_0
    else
        tmp = re * (0.5d0 * (re * sin(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -3.2e-8) {
		tmp = t_0;
	} else if (re <= 1.05e-6) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 5.2e+157) {
		tmp = t_0;
	} else {
		tmp = re * (0.5 * (re * Math.sin(im)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -3.2e-8:
		tmp = t_0
	elif re <= 1.05e-6:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 5.2e+157:
		tmp = t_0
	else:
		tmp = re * (0.5 * (re * math.sin(im)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -3.2e-8)
		tmp = t_0;
	elseif (re <= 1.05e-6)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 5.2e+157)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(0.5 * Float64(re * sin(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -3.2e-8)
		tmp = t_0;
	elseif (re <= 1.05e-6)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 5.2e+157)
		tmp = t_0;
	else
		tmp = re * (0.5 * (re * sin(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -3.2e-8], t$95$0, If[LessEqual[re, 1.05e-6], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.2e+157], t$95$0, N[(re * N[(0.5 * N[(re * N[Sin[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.2000000000000002e-8 or 1.0499999999999999e-6 < re < 5.20000000000000022e157

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.3%

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

   if -3.2000000000000002e-8 < re < 1.0499999999999999e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   if 5.20000000000000022e157 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. associate-*r* 91.6%

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

      7. *-commutative 91.6%

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

      8. associate-*l* 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(re \cdot \sin im\right)\right)\\ \end{array} \]

Alternative 6: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-8} \lor \neg \left(re \leq 1.05 \cdot 10^{-6}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -3.2e-8) (not (<= re 1.05e-6)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -3.2e-8) || !(re <= 1.05e-6)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-3.2d-8)) .or. (.not. (re <= 1.05d-6))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -3.2e-8) || !(re <= 1.05e-6)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -3.2e-8) or not (re <= 1.05e-6):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -3.2e-8) || !(re <= 1.05e-6))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -3.2e-8) || ~((re <= 1.05e-6)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -3.2e-8], N[Not[LessEqual[re, 1.05e-6]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.2000000000000002e-8 or 1.0499999999999999e-6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.9%

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

   if -3.2000000000000002e-8 < re < 1.0499999999999999e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-8} \lor \neg \left(re \leq 1.05 \cdot 10^{-6}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 7: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{t_0 - re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re 2.9e-9)
     (sin im)
     (if (<= re 5e+153)
       (+
        im
        (* im (/ (- (* 0.25 (* (* re re) (* re re))) (* re re)) (- t_0 re))))
       (* im t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= 2.9e-9) {
		tmp = sin(im);
	} else if (re <= 5e+153) {
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)));
	} else {
		tmp = 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 = 0.5d0 * (re * re)
    if (re <= 2.9d-9) then
        tmp = sin(im)
    else if (re <= 5d+153) then
        tmp = im + (im * (((0.25d0 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)))
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= 2.9e-9) {
		tmp = Math.sin(im);
	} else if (re <= 5e+153) {
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)));
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= 2.9e-9:
		tmp = math.sin(im)
	elif re <= 5e+153:
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)))
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= 2.9e-9)
		tmp = sin(im);
	elseif (re <= 5e+153)
		tmp = Float64(im + Float64(im * Float64(Float64(Float64(0.25 * Float64(Float64(re * re) * Float64(re * re))) - Float64(re * re)) / Float64(t_0 - re))));
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= 2.9e-9)
		tmp = sin(im);
	elseif (re <= 5e+153)
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)));
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 2.9e-9], N[Sin[im], $MachinePrecision], If[LessEqual[re, 5e+153], N[(im + N[(im * N[(N[(N[(0.25 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.89999999999999991e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 72.4%

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

   if 2.89999999999999991e-9 < re < 5.00000000000000018e153

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 76.9%

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

   3. Taylor expanded in re around 0 18.7%

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

      1. associate-+r+ 18.7%

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

      2. +-commutative 18.7%

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

      3. unpow2 18.7%

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

      4. associate-*r* 18.7%

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

      5. *-commutative 18.7%

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

      6. associate-*r* 18.7%

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

      7. distribute-rgt-out 18.7%

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

      8. +-commutative 18.7%

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

      9. associate-*r* 18.7%

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

      10. *-commutative 18.7%

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

      11. fma-def 18.7%

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

   5. Simplified 18.7%

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

      1. fma-udef 18.7%

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

      2. flip-+ 38.3%

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

   7. Applied egg-rr 38.3%

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

      1. unpow2 38.3%

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

      2. unpow2 38.3%

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

      3. swap-sqr 38.3%

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

      4. metadata-eval 38.3%

         \[\leadsto im + im \cdot \frac{\color{blue}{0.25} \cdot \left({re}^{2} \cdot {re}^{2}\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re} \]

      5. unpow2 38.3%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {re}^{2}\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re} \]

      6. unpow2 38.3%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot re\right)}\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re} \]

      7. unpow2 38.3%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{0.5 \cdot \color{blue}{{re}^{2}} - re} \]

      8. *-commutative 38.3%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{\color{blue}{{re}^{2} \cdot 0.5} - re} \]

      9. unpow2 38.3%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{\color{blue}{\left(re \cdot re\right)} \cdot 0.5 - re} \]

   9. Simplified 38.3%

      \[\leadsto im + im \cdot \color{blue}{\frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{\left(re \cdot re\right) \cdot 0.5 - re}} \]

   if 5.00000000000000018e153 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.5%

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

      1. associate-+r+ 97.5%

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

      2. +-commutative 97.5%

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

      3. *-commutative 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. *-commutative 97.5%

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

      6. associate-*r* 97.5%

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

      7. distribute-rgt-out 97.5%

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

      8. *-commutative 97.5%

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

      9. unpow2 97.5%

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

      10. associate-*l* 97.5%

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

   4. Simplified 97.5%

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

   5. Taylor expanded in re around inf 97.5%

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

      1. unpow2 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

      5. associate-*r* 97.5%

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

      6. associate-*r* 89.3%

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

      7. *-commutative 89.3%

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

      8. associate-*l* 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in im around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. *-commutative 70.6%

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

      3. associate-*r* 70.6%

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

      4. unpow2 70.6%

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

   10. Simplified 70.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 8: 40.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{t_0 - re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re 5e+153)
     (+
      im
      (* im (/ (- (* 0.25 (* (* re re) (* re re))) (* re re)) (- t_0 re))))
     (* im t_0))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= 5e+153) {
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)));
	} else {
		tmp = 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 = 0.5d0 * (re * re)
    if (re <= 5d+153) then
        tmp = im + (im * (((0.25d0 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)))
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= 5e+153) {
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)));
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= 5e+153:
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)))
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= 5e+153)
		tmp = Float64(im + Float64(im * Float64(Float64(Float64(0.25 * Float64(Float64(re * re) * Float64(re * re))) - Float64(re * re)) / Float64(t_0 - re))));
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= 5e+153)
		tmp = im + (im * (((0.25 * ((re * re) * (re * re))) - (re * re)) / (t_0 - re)));
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 5e+153], N[(im + N[(im * N[(N[(N[(0.25 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < 5.00000000000000018e153

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.0%

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

   3. Taylor expanded in re around 0 36.2%

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

      1. associate-+r+ 36.2%

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

      2. +-commutative 36.2%

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

      3. unpow2 36.2%

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

      4. associate-*r* 36.2%

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

      5. *-commutative 36.2%

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

      6. associate-*r* 36.2%

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

      7. distribute-rgt-out 36.3%

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

      8. +-commutative 36.3%

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

      9. associate-*r* 36.3%

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

      10. *-commutative 36.3%

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

      11. fma-def 36.3%

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

   5. Simplified 36.3%

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

      1. fma-udef 36.3%

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

      2. flip-+ 39.6%

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

   7. Applied egg-rr 39.6%

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

      1. unpow2 39.6%

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

      2. unpow2 39.6%

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

      3. swap-sqr 39.6%

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

      4. metadata-eval 39.6%

         \[\leadsto im + im \cdot \frac{\color{blue}{0.25} \cdot \left({re}^{2} \cdot {re}^{2}\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re} \]

      5. unpow2 39.6%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {re}^{2}\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re} \]

      6. unpow2 39.6%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot re\right)}\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re} \]

      7. unpow2 39.6%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{0.5 \cdot \color{blue}{{re}^{2}} - re} \]

      8. *-commutative 39.6%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{\color{blue}{{re}^{2} \cdot 0.5} - re} \]

      9. unpow2 39.6%

         \[\leadsto im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{\color{blue}{\left(re \cdot re\right)} \cdot 0.5 - re} \]

   9. Simplified 39.6%

      \[\leadsto im + im \cdot \color{blue}{\frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{\left(re \cdot re\right) \cdot 0.5 - re}} \]

   if 5.00000000000000018e153 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.5%

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

      1. associate-+r+ 97.5%

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

      2. +-commutative 97.5%

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

      3. *-commutative 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. *-commutative 97.5%

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

      6. associate-*r* 97.5%

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

      7. distribute-rgt-out 97.5%

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

      8. *-commutative 97.5%

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

      9. unpow2 97.5%

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

      10. associate-*l* 97.5%

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

   4. Simplified 97.5%

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

   5. Taylor expanded in re around inf 97.5%

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

      1. unpow2 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

      5. associate-*r* 97.5%

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

      6. associate-*r* 89.3%

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

      7. *-commutative 89.3%

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

      8. associate-*l* 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in im around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. *-commutative 70.6%

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

      3. associate-*r* 70.6%

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

      4. unpow2 70.6%

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

   10. Simplified 70.6%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \frac{0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) - re \cdot re}{0.5 \cdot \left(re \cdot re\right) - re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 9: 37.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ im + im \cdot \left(re + 0.5 \cdot \left(re \cdot re\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ im (* im (+ re (* 0.5 (* re re))))))

C

double code(double re, double im) {
	return im + (im * (re + (0.5 * (re * re))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im + (im * (re + (0.5d0 * (re * re))))
end function

Java

public static double code(double re, double im) {
	return im + (im * (re + (0.5 * (re * re))));
}

Python

def code(re, im):
	return im + (im * (re + (0.5 * (re * re))))

Julia

function code(re, im)
	return Float64(im + Float64(im * Float64(re + Float64(0.5 * Float64(re * re)))))
end

MATLAB

function tmp = code(re, im)
	tmp = im + (im * (re + (0.5 * (re * re))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 69.2%

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

3. Taylor expanded in re around 0 40.8%

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

   1. associate-+r+ 40.8%

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

   2. +-commutative 40.8%

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

   3. unpow2 40.8%

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

   4. associate-*r* 40.8%

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

   5. *-commutative 40.8%

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

   6. associate-*r* 40.8%

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

   7. distribute-rgt-out 40.9%

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

   8. +-commutative 40.9%

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

   9. associate-*r* 40.9%

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

   10. *-commutative 40.9%

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

   11. fma-def 40.9%

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

5. Simplified 40.9%

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

   1. fma-udef 40.9%

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

7. Applied egg-rr 40.9%

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

8. Final simplification 40.9%

   \[\leadsto im + im \cdot \left(re + 0.5 \cdot \left(re \cdot re\right)\right) \]

Alternative 10: 37.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.7) (+ im (* re im)) (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = im + (re * im);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.7d0) then
        tmp = im + (re * im)
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = im + (re * im);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7:
		tmp = im + (re * im)
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7)
		tmp = Float64(im + Float64(re * im));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7)
		tmp = im + (re * im);
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 67.9%

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

   3. Taylor expanded in re around 0 40.4%

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

   if 2.7000000000000002 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 49.5%

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

      1. associate-+r+ 49.5%

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

      2. +-commutative 49.5%

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

      3. *-commutative 49.5%

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

      4. distribute-lft1-in 49.5%

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

      5. *-commutative 49.5%

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

      6. associate-*r* 49.5%

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

      7. distribute-rgt-out 49.5%

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

      8. *-commutative 49.5%

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

      9. unpow2 49.5%

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

      10. associate-*l* 49.5%

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

   4. Simplified 49.5%

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

   5. Taylor expanded in re around inf 49.5%

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

      1. unpow2 49.5%

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

      2. *-commutative 49.5%

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

      3. associate-*r* 49.5%

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

      4. *-commutative 49.5%

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

      5. associate-*r* 49.5%

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

      6. associate-*r* 45.6%

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

      7. *-commutative 45.6%

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

      8. associate-*l* 45.6%

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

   7. Simplified 45.6%

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

   8. Taylor expanded in im around 0 41.5%

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

      1. *-commutative 41.5%

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

      2. *-commutative 41.5%

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

      3. associate-*r* 41.5%

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

      4. unpow2 41.5%

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

   10. Simplified 41.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 11: 30.3% 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 67.9%

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

   3. Taylor expanded in re around 0 39.7%

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

   if 1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.9%

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

   3. Taylor expanded in re around 0 15.6%

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

   4. Taylor expanded in re around inf 15.6%

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

4. Final simplification 33.1%

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

Alternative 12: 30.2% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 69.2%

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

3. Taylor expanded in re around 0 33.6%

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

4. Final simplification 33.6%

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

Alternative 13: 27.6% 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 69.2%

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

3. Taylor expanded in re around 0 29.6%

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

4. Final simplification 29.6%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023189 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))