math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.0×

• 25.3% 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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]
4. Add Preprocessing

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-137} \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1e-137) (not (<= (exp re) 2.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1e-137) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(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 ((exp(re) <= 1d-137) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1e-137) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1e-137) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1e-137) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1e-137) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1e-137], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 9.99999999999999978e-138 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 81.8%

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

   if 9.99999999999999978e-138 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.8%

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

      1. distribute-rgt1-in 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-137} \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im)
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $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. Add Preprocessing

   3. Taylor expanded in im around 0 82.4%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -320 \lor \neg \left(re \leq 1.25 \cdot 10^{+20}\right) \land re \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -320.0) (and (not (<= re 1.25e+20)) (<= re 1.02e+149)))
   (exp re)
   (* (cos im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -320.0) || (!(re <= 1.25e+20) && (re <= 1.02e+149))) {
		tmp = exp(re);
	} else {
		tmp = cos(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 <= (-320.0d0)) .or. (.not. (re <= 1.25d+20)) .and. (re <= 1.02d+149)) then
        tmp = exp(re)
    else
        tmp = cos(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 <= -320.0) || (!(re <= 1.25e+20) && (re <= 1.02e+149))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -320.0) or (not (re <= 1.25e+20) and (re <= 1.02e+149)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -320.0) || (!(re <= 1.25e+20) && (re <= 1.02e+149)))
		tmp = exp(re);
	else
		tmp = Float64(cos(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 <= -320.0) || (~((re <= 1.25e+20)) && (re <= 1.02e+149)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -320.0], And[N[Not[LessEqual[re, 1.25e+20]], $MachinePrecision], LessEqual[re, 1.02e+149]]], N[Exp[re], $MachinePrecision], N[(N[Cos[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 < -320 or 1.25e20 < re < 1.01999999999999997e149

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.7%

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

   if -320 < re < 1.25e20 or 1.01999999999999997e149 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 99.5%

         \[\leadsto e^{re} \cdot \color{blue}{\log \left(e^{\cos im}\right)} \]

      2. add-sqr-sqrt 99.3%

         \[\leadsto e^{re} \cdot \log \color{blue}{\left(\sqrt{e^{\cos im}} \cdot \sqrt{e^{\cos im}}\right)} \]

      3. log-prod 99.3%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos im}}\right) + \log \left(\sqrt{e^{\cos im}}\right)\right)} \]

   4. Applied egg-rr 99.3%

      \[\leadsto e^{re} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos im}}\right) + \log \left(\sqrt{e^{\cos im}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 99.3%

         \[\leadsto e^{re} \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos im}}\right)\right)} \]

   6. Simplified 99.3%

      \[\leadsto e^{re} \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos im}}\right)\right)} \]
   7. Step-by-step derivation

      1. pow1/2 99.3%

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

      2. pow-exp 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in re around 0 97.7%

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

       1. distribute-lft-in 97.7%

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

       2. *-commutative 97.7%

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

       3. associate-+r+ 97.7%

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

       4. *-commutative 97.7%

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

       5. distribute-rgt1-in 97.7%

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

       6. associate-*r* 97.7%

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

       7. associate-*r* 97.7%

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

       8. distribute-rgt-out 97.7%

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

   11. Simplified 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -320 \lor \neg \left(re \leq 1.25 \cdot 10^{+20}\right) \land re \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6e+20) (cos im) (+ (+ re 1.0) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+20) {
		tmp = cos(im);
	} else {
		tmp = (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 <= 1.6d+20) then
        tmp = cos(im)
    else
        tmp = (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 <= 1.6e+20) {
		tmp = Math.cos(im);
	} else {
		tmp = (re + 1.0) + (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6e+20:
		tmp = math.cos(im)
	else:
		tmp = (re + 1.0) + (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6e+20)
		tmp = cos(im);
	else
		tmp = 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 <= 1.6e+20)
		tmp = cos(im);
	else
		tmp = (re + 1.0) + (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6e+20], N[Cos[im], $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.6e20

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 67.1%

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

   if 1.6e20 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto e^{re} \cdot \color{blue}{\log \left(e^{\cos im}\right)} \]

      2. add-sqr-sqrt 100.0%

         \[\leadsto e^{re} \cdot \log \color{blue}{\left(\sqrt{e^{\cos im}} \cdot \sqrt{e^{\cos im}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos im}}\right) + \log \left(\sqrt{e^{\cos im}}\right)\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos im}}\right) + \log \left(\sqrt{e^{\cos im}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 100.0%

         \[\leadsto e^{re} \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos im}}\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos im}}\right)\right)} \]
   7. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. pow-exp 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in re around 0 52.6%

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

       1. distribute-lft-in 52.6%

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

       2. *-commutative 52.6%

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

       3. associate-+r+ 52.6%

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

       4. *-commutative 52.6%

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

       5. distribute-rgt1-in 52.6%

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

       6. associate-*r* 52.6%

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

       7. associate-*r* 52.6%

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

       8. distribute-rgt-out 52.6%

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

   11. Simplified 52.6%

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

   12. Taylor expanded in im around 0 27.1%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.4% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 99.7%

      \[\leadsto e^{re} \cdot \color{blue}{\log \left(e^{\cos im}\right)} \]

   2. add-sqr-sqrt 99.5%

      \[\leadsto e^{re} \cdot \log \color{blue}{\left(\sqrt{e^{\cos im}} \cdot \sqrt{e^{\cos im}}\right)} \]

   3. log-prod 99.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos im}}\right) + \log \left(\sqrt{e^{\cos im}}\right)\right)} \]

4. Applied egg-rr 99.6%

   \[\leadsto e^{re} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos im}}\right) + \log \left(\sqrt{e^{\cos im}}\right)\right)} \]
5. Step-by-step derivation

   1. count-2 99.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos im}}\right)\right)} \]

6. Simplified 99.6%

   \[\leadsto e^{re} \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos im}}\right)\right)} \]
7. Step-by-step derivation

   1. pow1/2 99.6%

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

   2. pow-exp 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in re around 0 63.6%

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

    1. distribute-lft-in 63.6%

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

    2. *-commutative 63.6%

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

    3. associate-+r+ 63.6%

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

    4. *-commutative 63.6%

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

    5. distribute-rgt1-in 63.6%

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

    6. associate-*r* 63.6%

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

    7. associate-*r* 63.6%

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

    8. distribute-rgt-out 63.6%

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

11. Simplified 63.6%

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

12. Taylor expanded in im around 0 36.9%

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

13. Final simplification 36.9%

    \[\leadsto \left(re + 1\right) + re \cdot \left(re \cdot 0.5\right) \]
14. Add Preprocessing

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in re around 0 52.5%

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

   1. distribute-rgt1-in 52.5%

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

5. Simplified 52.5%

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

6. Taylor expanded in im around 0 31.4%

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

7. Final simplification 31.4%

   \[\leadsto re + 1 \]
8. Add Preprocessing

Alternative 8: 28.0% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 52.1%

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

4. Taylor expanded in im around 0 31.3%

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

5. Final simplification 31.3%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))