math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 10

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 2.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.7%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. distribute-lft-in 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0) (exp re) (if (<= (exp re) 2.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.7%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 97.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.028 \lor \neg \left(re \leq 0.009\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.028) (and (not (<= re 0.009)) (<= re 1.05e+103)))
   (exp re)
   (*
    (cos im)
    (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.028) || (!(re <= 0.009) && (re <= 1.05e+103))) {
		tmp = exp(re);
	} else {
		tmp = cos(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) :: tmp
    if ((re <= (-0.028d0)) .or. (.not. (re <= 0.009d0)) .and. (re <= 1.05d+103)) then
        tmp = exp(re)
    else
        tmp = cos(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 tmp;
	if ((re <= -0.028) || (!(re <= 0.009) && (re <= 1.05e+103))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.028) or (not (re <= 0.009) and (re <= 1.05e+103)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.028) || (!(re <= 0.009) && (re <= 1.05e+103)))
		tmp = exp(re);
	else
		tmp = Float64(cos(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)
	tmp = 0.0;
	if ((re <= -0.028) || (~((re <= 0.009)) && (re <= 1.05e+103)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.028], And[N[Not[LessEqual[re, 0.009]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[Exp[re], $MachinePrecision], N[(N[Cos[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 2 regimes

2. if re < -0.0280000000000000006 or 0.00899999999999999932 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.9%

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

   if -0.0280000000000000006 < re < 0.00899999999999999932 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-lft1-in 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. +-commutative 100.0%

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

      12. cube-mult 100.0%

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

      13. unpow2 100.0%

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

      14. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.028 \lor \neg \left(re \leq 0.009\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 5: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.028:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0029:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.028)
   (exp re)
   (if (<= re 0.0029) (* (cos im) (+ (+ re 1.0) (* re (* re 0.5)))) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.028) {
		tmp = exp(re);
	} else if (re <= 0.0029) {
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else {
		tmp = exp(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 <= (-0.028d0)) then
        tmp = exp(re)
    else if (re <= 0.0029d0) then
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.028) {
		tmp = Math.exp(re);
	} else if (re <= 0.0029) {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.028:
		tmp = math.exp(re)
	elif re <= 0.0029:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.028)
		tmp = exp(re);
	elseif (re <= 0.0029)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.028)
		tmp = exp(re);
	elseif (re <= 0.0029)
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.028], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0029], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0280000000000000006 or 0.0029 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.7%

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

   if -0.0280000000000000006 < re < 0.0029

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-lft1-in 99.9%

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

      5. distribute-rgt-out 99.8%

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

      6. +-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. unpow2 99.8%

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

      9. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.028:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0029:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.8e+23) (cos im) (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+23) {
		tmp = cos(im);
	} else {
		tmp = re * (1.0 + (-0.5 * (im * 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 <= 4.8d+23) then
        tmp = cos(im)
    else
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+23) {
		tmp = Math.cos(im);
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.8e+23:
		tmp = math.cos(im)
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.8e+23)
		tmp = cos(im);
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.8e+23)
		tmp = cos(im);
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.8e+23], N[Cos[im], $MachinePrecision], N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.8e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 60.0%

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

   if 4.8e23 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.6%

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

      1. *-rgt-identity 5.6%

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

      2. distribute-lft-in 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in im around 0 19.9%

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

      1. associate-+r+ 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

      4. *-commutative 19.9%

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

      5. unpow2 19.9%

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

   7. Simplified 19.9%

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

   8. Taylor expanded in re around inf 19.9%

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

      1. *-commutative 19.9%

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

      2. unpow2 19.9%

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

   10. Simplified 19.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 33.0% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.8e+21) (+ re 1.0) (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+21) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * 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 <= 4.8d+21) then
        tmp = re + 1.0d0
    else
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+21) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.8e+21:
		tmp = re + 1.0
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.8e+21)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.8e+21)
		tmp = re + 1.0;
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.8e+21], N[(re + 1.0), $MachinePrecision], N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.8e21

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 60.8%

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

      1. *-rgt-identity 60.8%

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

      2. distribute-lft-in 60.8%

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

   4. Simplified 60.8%

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

   5. Taylor expanded in im around 0 35.2%

      \[\leadsto \color{blue}{1 + re} \]
   6. Step-by-step derivation

      1. +-commutative 35.2%

         \[\leadsto \color{blue}{re + 1} \]

   7. Simplified 35.2%

      \[\leadsto \color{blue}{re + 1} \]

   if 4.8e21 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.6%

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

      1. *-rgt-identity 5.6%

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

      2. distribute-lft-in 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in im around 0 19.9%

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

      1. associate-+r+ 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

      4. *-commutative 19.9%

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

      5. unpow2 19.9%

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

   7. Simplified 19.9%

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

   8. Taylor expanded in re around inf 19.9%

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

      1. *-commutative 19.9%

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

      2. unpow2 19.9%

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

   10. Simplified 19.9%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 31.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \left(re + 1\right) + \left(im \cdot -0.5\right) \cdot \left(re \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ (+ re 1.0) (* (* im -0.5) (* re im))))

C

double code(double re, double im) {
	return (re + 1.0) + ((im * -0.5) * (re * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (re + 1.0d0) + ((im * (-0.5d0)) * (re * im))
end function

Java

public static double code(double re, double im) {
	return (re + 1.0) + ((im * -0.5) * (re * im));
}

Python

def code(re, im):
	return (re + 1.0) + ((im * -0.5) * (re * im))

Julia

function code(re, im)
	return Float64(Float64(re + 1.0) + Float64(Float64(im * -0.5) * Float64(re * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (re + 1.0) + ((im * -0.5) * (re * im));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 48.5%

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

   1. *-rgt-identity 48.5%

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

   2. distribute-lft-in 48.5%

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

4. Simplified 48.5%

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

5. Taylor expanded in im around 0 30.3%

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

   1. associate-+r+ 30.3%

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

   2. +-commutative 30.3%

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

   3. +-commutative 30.3%

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

   4. *-commutative 30.3%

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

   5. unpow2 30.3%

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

7. Simplified 30.3%

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

8. Taylor expanded in re around inf 30.5%

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

   1. *-commutative 30.5%

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

   2. unpow2 30.5%

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

   3. associate-*l* 30.5%

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

   4. associate-*r* 30.5%

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

   5. *-commutative 30.5%

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

10. Simplified 30.5%

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

11. Final simplification 30.5%

    \[\leadsto \left(re + 1\right) + \left(im \cdot -0.5\right) \cdot \left(re \cdot im\right) \]

Alternative 9: 28.9% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \left(im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 (* -0.5 (* im im))))

C

double code(double re, double im) {
	return 1.0 + (-0.5 * (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + ((-0.5d0) * (im * im))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (-0.5 * (im * im));
}

Python

def code(re, im):
	return 1.0 + (-0.5 * (im * im))

Julia

function code(re, im)
	return Float64(1.0 + Float64(-0.5 * Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (-0.5 * (im * im));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 48.5%

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

   1. *-rgt-identity 48.5%

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

   2. distribute-lft-in 48.5%

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

4. Simplified 48.5%

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

5. Taylor expanded in im around 0 30.3%

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

   1. associate-+r+ 30.3%

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

   2. +-commutative 30.3%

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

   3. +-commutative 30.3%

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

   4. *-commutative 30.3%

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

   5. unpow2 30.3%

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

7. Simplified 30.3%

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

8. Taylor expanded in re around 0 29.2%

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

   1. unpow2 29.2%

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

10. Simplified 29.2%

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

11. Final simplification 29.2%

    \[\leadsto 1 + -0.5 \cdot \left(im \cdot im\right) \]

Alternative 10: 28.4% accurate, 67.7× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 48.5%

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

   1. *-rgt-identity 48.5%

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

   2. distribute-lft-in 48.5%

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

4. Simplified 48.5%

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

5. Taylor expanded in im around 0 28.4%

   \[\leadsto \color{blue}{1 + re} \]
6. Step-by-step derivation

   1. +-commutative 28.4%

      \[\leadsto \color{blue}{re + 1} \]

7. Simplified 28.4%

   \[\leadsto \color{blue}{re + 1} \]

8. Final simplification 28.4%

   \[\leadsto re + 1 \]

Reproduce

herbie shell --seed 2023172 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))