math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 12

Speedup: 1.0×

• 24.0% 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: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.22 \lor \neg \left(re \leq 600000\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.22) (and (not (<= re 600000.0)) (<= re 1e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ (+ re 1.0) (* (* re re) (+ (* re 0.16666666666666666) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.22) || (!(re <= 600000.0) && (re <= 1e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.22d0)) .or. (.not. (re <= 600000.0d0)) .and. (re <= 1d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.22) || (!(re <= 600000.0) && (re <= 1e+103))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.22) or (not (re <= 600000.0) and (re <= 1e+103)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.22) || (!(re <= 600000.0) && (re <= 1e+103)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.22) || (~((re <= 600000.0)) && (re <= 1e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.22], And[N[Not[LessEqual[re, 600000.0]], $MachinePrecision], LessEqual[re, 1e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.220000000000000001 or 6e5 < re < 1e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.0%

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

   if -0.220000000000000001 < re < 6e5 or 1e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. associate-+r+ 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt1-in 99.4%

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

      4. *-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*r* 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. distribute-rgt-out 99.4%

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

      11. distribute-lft-out 99.4%

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

      12. +-commutative 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.22 \lor \neg \left(re \leq 600000\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \]

Alternative 3: 93.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.02 \lor \neg \left(re \leq 600000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.02) (not (<= re 600000.0)))
   (* (exp re) im)
   (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.02) || !(re <= 600000.0)) {
		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 <= (-0.02d0)) .or. (.not. (re <= 600000.0d0))) 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 <= -0.02) || !(re <= 600000.0)) {
		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 <= -0.02) or not (re <= 600000.0):
		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 <= -0.02) || !(re <= 600000.0))
		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 <= -0.02) || ~((re <= 600000.0)))
		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, -0.02], N[Not[LessEqual[re, 600000.0]], $MachinePrecision]], 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 < -0.0200000000000000004 or 6e5 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.4%

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

   if -0.0200000000000000004 < re < 6e5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-lft1-in 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.2%

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

      7. distribute-rgt-out 99.2%

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

      8. *-commutative 99.2%

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

      9. unpow2 99.2%

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

      10. associate-*l* 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.02 \lor \neg \left(re \leq 600000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00027 \lor \neg \left(re \leq 600000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00027) (not (<= re 600000.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00027) || !(re <= 600000.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00027d0)) .or. (.not. (re <= 600000.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00027) || !(re <= 600000.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00027) or not (re <= 600000.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00027) || !(re <= 600000.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00027) || ~((re <= 600000.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00027], N[Not[LessEqual[re, 600000.0]], $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 < -2.70000000000000003e-4 or 6e5 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.4%

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

   if -2.70000000000000003e-4 < re < 6e5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. distribute-rgt1-in 98.9%

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

   4. Simplified 98.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00027 \lor \neg \left(re \leq 600000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 5: 92.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-5} \lor \neg \left(re \leq 600000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5e-5) (not (<= re 600000.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -5e-5) || !(re <= 600000.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-5d-5)) .or. (.not. (re <= 600000.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -5e-5) || !(re <= 600000.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -5e-5) or not (re <= 600000.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -5e-5) || !(re <= 600000.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -5e-5) || ~((re <= 600000.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -5e-5], N[Not[LessEqual[re, 600000.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.00000000000000024e-5 or 6e5 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.4%

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

   if -5.00000000000000024e-5 < re < 6e5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-5} \lor \neg \left(re \leq 600000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 6: 37.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{expm1}\left(im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+61) (expm1 im) (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+61) {
		tmp = expm1(im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+61) {
		tmp = Math.expm1(im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+61:
		tmp = math.expm1(im)
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+61)
		tmp = expm1(im);
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+61], N[(Exp[im] - 1), $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.5e61

   1. Initial program 100.0%

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

      1. expm1-log1p-u 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in re around 0 52.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\sin im + 1\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 52.9%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(1 + \sin im\right)}\right) \]

      2. log1p-def 63.1%

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

   6. Simplified 63.1%

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

   7. Taylor expanded in im around 0 30.2%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{im}\right) \]

   if 1.5e61 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 48.5%

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

      1. associate-+r+ 48.5%

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

      2. +-commutative 48.5%

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

      3. *-commutative 48.5%

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

      4. distribute-lft1-in 48.5%

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

      5. *-commutative 48.5%

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

      6. associate-*r* 48.5%

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

      7. distribute-rgt-out 48.5%

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

      8. *-commutative 48.5%

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

      9. unpow2 48.5%

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

      10. associate-*l* 48.5%

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

   4. Simplified 48.5%

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

   5. Taylor expanded in re around inf 48.5%

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

      1. unpow2 48.5%

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

      2. *-commutative 48.5%

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

      3. associate-*r* 48.5%

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

      4. associate-*r* 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in im around 0 49.2%

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

      1. unpow2 49.2%

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

      2. associate-*r* 49.2%

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

      3. *-commutative 49.2%

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

      4. associate-*r* 49.2%

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

   10. Simplified 49.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{expm1}\left(im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e+67) (sin im) (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e+67) {
		tmp = sin(im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.9d+67) then
        tmp = sin(im)
    else
        tmp = im * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.9e+67) {
		tmp = Math.sin(im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e+67:
		tmp = math.sin(im)
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e+67)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9e+67)
		tmp = sin(im);
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e+67], N[Sin[im], $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.9000000000000001e67

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 62.6%

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

   if 1.9000000000000001e67 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 50.6%

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

      1. associate-+r+ 50.6%

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

      2. +-commutative 50.6%

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

      3. *-commutative 50.6%

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

      4. distribute-lft1-in 50.6%

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

      5. *-commutative 50.6%

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

      6. associate-*r* 50.6%

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

      7. distribute-rgt-out 50.6%

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

      8. *-commutative 50.6%

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

      9. unpow2 50.6%

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

      10. associate-*l* 50.6%

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

   4. Simplified 50.6%

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

   5. Taylor expanded in re around inf 50.6%

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

      1. unpow2 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*r* 50.6%

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

      4. associate-*r* 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in im around 0 51.5%

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

      1. unpow2 51.5%

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

      2. associate-*r* 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-*r* 51.5%

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

   10. Simplified 51.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 33.5% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.15e-9) im (* re (* im (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.15e-9) {
		tmp = im;
	} else {
		tmp = re * (im * (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 <= 1.15d-9) then
        tmp = im
    else
        tmp = re * (im * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.15e-9) {
		tmp = im;
	} else {
		tmp = re * (im * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.15e-9:
		tmp = im
	else:
		tmp = re * (im * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.15e-9)
		tmp = im;
	else
		tmp = Float64(re * Float64(im * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.15e-9)
		tmp = im;
	else
		tmp = re * (im * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.15e-9], im, N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.15e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. distribute-rgt1-in 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in im around 0 30.6%

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

   6. Taylor expanded in re around 0 31.0%

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

   if 1.15e-9 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 40.1%

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

      1. associate-+r+ 40.1%

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

      2. +-commutative 40.1%

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

      3. *-commutative 40.1%

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

      4. distribute-lft1-in 40.1%

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

      5. *-commutative 40.1%

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

      6. associate-*r* 40.1%

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

      7. distribute-rgt-out 40.1%

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

      8. *-commutative 40.1%

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

      9. unpow2 40.1%

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

      10. associate-*l* 40.1%

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

   4. Simplified 40.1%

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

   5. Taylor expanded in re around inf 37.2%

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

      1. unpow2 37.2%

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

      2. *-commutative 37.2%

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

      3. associate-*r* 37.2%

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

      4. associate-*r* 37.2%

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

   7. Simplified 37.2%

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

   8. Taylor expanded in im around 0 37.2%

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

      1. unpow2 37.2%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. associate-*r* 37.2%

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

      5. associate-*l* 27.8%

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

   10. Simplified 27.8%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 36.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.15e-9) im (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.15e-9) {
		tmp = im;
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.15d-9) then
        tmp = im
    else
        tmp = im * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.15e-9) {
		tmp = im;
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.15e-9:
		tmp = im
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.15e-9)
		tmp = im;
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.15e-9)
		tmp = im;
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.15e-9], im, N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.15e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. distribute-rgt1-in 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in im around 0 30.6%

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

   6. Taylor expanded in re around 0 31.0%

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

   if 1.15e-9 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 40.1%

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

      1. associate-+r+ 40.1%

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

      2. +-commutative 40.1%

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

      3. *-commutative 40.1%

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

      4. distribute-lft1-in 40.1%

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

      5. *-commutative 40.1%

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

      6. associate-*r* 40.1%

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

      7. distribute-rgt-out 40.1%

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

      8. *-commutative 40.1%

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

      9. unpow2 40.1%

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

      10. associate-*l* 40.1%

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

   4. Simplified 40.1%

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

   5. Taylor expanded in re around inf 37.2%

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

      1. unpow2 37.2%

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

      2. *-commutative 37.2%

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

      3. associate-*r* 37.2%

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

      4. associate-*r* 37.2%

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

   7. Simplified 37.2%

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

   8. Taylor expanded in im around 0 37.2%

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

      1. unpow2 37.2%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. associate-*r* 37.2%

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

   10. Simplified 37.2%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 29.2% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.15e-9) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.15e-9) {
		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 <= 1.15d-9) 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 <= 1.15e-9) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.15e-9:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.15e-9)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.15e-9)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.15e-9], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.15e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. distribute-rgt1-in 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in im around 0 30.6%

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

   6. Taylor expanded in re around 0 31.0%

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

   if 1.15e-9 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 6.5%

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

      1. *-commutative 6.5%

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

      2. distribute-rgt1-in 6.5%

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

   4. Simplified 6.5%

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

   5. Taylor expanded in im around 0 14.3%

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

   6. Taylor expanded in re around inf 14.3%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 11: 29.2% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. distribute-rgt1-in 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in im around 0 26.8%

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

6. Final simplification 26.8%

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

Alternative 12: 26.2% 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 re around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. distribute-rgt1-in 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in im around 0 26.8%

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

6. Taylor expanded in re around 0 24.4%

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

7. Final simplification 24.4%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023240 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))