math.exp on complex, real part

Percentage Accurate: 100.0% → 99.9%

Time: 8.1s

Alternatives: 13

Speedup: 1.0×

• 24.9% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left({e}^{\left(re \cdot 0.6666666666666666\right)} \cdot {e}^{\left(re \cdot 0.3333333333333333\right)}\right) \cdot \cos im \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* (pow E (* re 0.6666666666666666)) (pow E (* re 0.3333333333333333)))
  (cos im)))

C

double code(double re, double im) {
	return (pow(((double) M_E), (re * 0.6666666666666666)) * pow(((double) M_E), (re * 0.3333333333333333))) * cos(im);
}

Java

public static double code(double re, double im) {
	return (Math.pow(Math.E, (re * 0.6666666666666666)) * Math.pow(Math.E, (re * 0.3333333333333333))) * Math.cos(im);
}

Python

def code(re, im):
	return (math.pow(math.e, (re * 0.6666666666666666)) * math.pow(math.e, (re * 0.3333333333333333))) * math.cos(im)

Julia

function code(re, im)
	return Float64(Float64((exp(1) ^ Float64(re * 0.6666666666666666)) * (exp(1) ^ Float64(re * 0.3333333333333333))) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = ((2.71828182845904523536 ^ (re * 0.6666666666666666)) * (2.71828182845904523536 ^ (re * 0.3333333333333333))) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[(N[Power[E, N[(re * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] * N[Power[E, N[(re * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. add-log-exp 100.0%

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

   4. add-cube-cbrt 100.0%

      \[\leadsto {\left(e^{1}\right)}^{\log \color{blue}{\left(\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \sqrt[3]{e^{re}}\right)}} \cdot \cos im \]

   5. log-prod 100.0%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\log \left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) + \log \left(\sqrt[3]{e^{re}}\right)\right)}} \cdot \cos im \]

   6. unpow-prod-up 100.0%

      \[\leadsto \color{blue}{\left({\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right)} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right)} \cdot \cos im \]

   7. exp-1-e 100.0%

      \[\leadsto \left({\color{blue}{e}}^{\log \left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right)} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   8. pow2 100.0%

      \[\leadsto \left({e}^{\log \color{blue}{\left({\left(\sqrt[3]{e^{re}}\right)}^{2}\right)}} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   9. log-pow 100.0%

      \[\leadsto \left({e}^{\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{e^{re}}\right)\right)}} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   10. pow1/3 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \log \color{blue}{\left({\left(e^{re}\right)}^{0.3333333333333333}\right)}\right)} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   11. log-pow 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{re}\right)\right)}\right)} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   12. add-log-exp 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \left(0.3333333333333333 \cdot \color{blue}{re}\right)\right)} \cdot {\left(e^{1}\right)}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   13. exp-1-e 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \left(0.3333333333333333 \cdot re\right)\right)} \cdot {\color{blue}{e}}^{\log \left(\sqrt[3]{e^{re}}\right)}\right) \cdot \cos im \]

   14. pow1/3 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \left(0.3333333333333333 \cdot re\right)\right)} \cdot {e}^{\log \color{blue}{\left({\left(e^{re}\right)}^{0.3333333333333333}\right)}}\right) \cdot \cos im \]

   15. log-pow 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \left(0.3333333333333333 \cdot re\right)\right)} \cdot {e}^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{re}\right)\right)}}\right) \cdot \cos im \]

   16. add-log-exp 100.0%

       \[\leadsto \left({e}^{\left(2 \cdot \left(0.3333333333333333 \cdot re\right)\right)} \cdot {e}^{\left(0.3333333333333333 \cdot \color{blue}{re}\right)}\right) \cdot \cos im \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left({e}^{\left(2 \cdot \left(0.3333333333333333 \cdot re\right)\right)} \cdot {e}^{\left(0.3333333333333333 \cdot re\right)}\right)} \cdot \cos im \]
5. Step-by-step derivation

   1. associate-*r* 100.0%

      \[\leadsto \left({e}^{\color{blue}{\left(\left(2 \cdot 0.3333333333333333\right) \cdot re\right)}} \cdot {e}^{\left(0.3333333333333333 \cdot re\right)}\right) \cdot \cos im \]

   2. *-commutative 100.0%

      \[\leadsto \left({e}^{\color{blue}{\left(re \cdot \left(2 \cdot 0.3333333333333333\right)\right)}} \cdot {e}^{\left(0.3333333333333333 \cdot re\right)}\right) \cdot \cos im \]

   3. metadata-eval 100.0%

      \[\leadsto \left({e}^{\left(re \cdot \color{blue}{0.6666666666666666}\right)} \cdot {e}^{\left(0.3333333333333333 \cdot re\right)}\right) \cdot \cos im \]

   4. *-commutative 100.0%

      \[\leadsto \left({e}^{\left(re \cdot 0.6666666666666666\right)} \cdot {e}^{\color{blue}{\left(re \cdot 0.3333333333333333\right)}}\right) \cdot \cos im \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\left({e}^{\left(re \cdot 0.6666666666666666\right)} \cdot {e}^{\left(re \cdot 0.3333333333333333\right)}\right)} \cdot \cos im \]
7. Add Preprocessing

Alternative 2: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\left(--1\right) - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0)))
   (exp re)
   (/ (cos im) (- (- -1.0) re))))

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) / (-(-1.0) - 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) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) / (-(-1.0d0) - re)
    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) / (-(-1.0) - re);
	}
	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) / (-(-1.0) - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) / Float64(Float64(-(-1.0)) - re));
	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) / (-(-1.0) - re);
	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[(N[Cos[im], $MachinePrecision] / N[((--1.0) - re), $MachinePrecision]), $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 93.8%

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

   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 99.5%

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

      1. distribute-rgt1-in 99.5%

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

   5. Simplified 99.5%

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

      1. flip-+ 99.5%

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

      2. associate-*l/ 99.5%

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

      3. metadata-eval 99.5%

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

      4. fmm-def 99.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(re, re, -1\right)} \cdot \cos im}{re - 1} \]

      5. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(re, re, \color{blue}{-1}\right) \cdot \cos im}{re - 1} \]

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in re around 0 99.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re + -1} \]
   9. Step-by-step derivation

      1. mul-1-neg 99.5%

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

   10. Simplified 99.5%

       \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\left(--1\right) - re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

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) * (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) <= 0.0d0) .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) <= 0.0) || !(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) <= 0.0) 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) <= 0.0) || !(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) <= 0.0) || ~((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], 0.0], 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) < 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 93.8%

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

   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 99.5%

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

      1. distribute-rgt1-in 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 96.6%

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

Alternative 4: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999912773052766 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999912773052766) (not (<= (exp re) 2.0)))
   (exp re)
   (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999912773052766) || !(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) <= 0.9999912773052766d0) .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) <= 0.9999912773052766) || !(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) <= 0.9999912773052766) 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) <= 0.9999912773052766) || !(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) <= 0.9999912773052766) || ~((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], 0.9999912773052766], 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) < 0.99999127730527659 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 93.2%

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

   if 0.99999127730527659 < (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%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999912773052766 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 6: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0086 \lor \neg \left(re \leq 0.115\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0086) (and (not (<= re 0.115)) (<= re 1.02e+103)))
   (exp re)
   (*
    (cos im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0086) || (!(re <= 0.115) && (re <= 1.02e+103))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (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 <= (-0.0086d0)) .or. (.not. (re <= 0.115d0)) .and. (re <= 1.02d+103)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (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 <= -0.0086) || (!(re <= 0.115) && (re <= 1.02e+103))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0086) or (not (re <= 0.115) and (re <= 1.02e+103)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0086) || (!(re <= 0.115) && (re <= 1.02e+103)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0086) || (~((re <= 0.115)) && (re <= 1.02e+103)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0086], And[N[Not[LessEqual[re, 0.115]], $MachinePrecision], LessEqual[re, 1.02e+103]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0086 or 0.115000000000000005 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.5%

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

   if -0.0086 < re < 0.115000000000000005 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0086 \lor \neg \left(re \leq 0.115\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0073 \lor \neg \left(re \leq 0.00136\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0073) (not (<= re 0.00136)))
   (exp re)
   (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0073) || !(re <= 0.00136)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0073d0)) .or. (.not. (re <= 0.00136d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0073) || !(re <= 0.00136)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0073) or not (re <= 0.00136):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0073) || !(re <= 0.00136))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0073) || ~((re <= 0.00136)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0073], N[Not[LessEqual[re, 0.00136]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00730000000000000007 or 0.00136 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.8%

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

   if -0.00730000000000000007 < re < 0.00136

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0073 \lor \neg \left(re \leq 0.00136\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+27}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e+27)
   (cos im)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e+27) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (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 <= 1.9d+27) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (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 <= 1.9e+27) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e+27:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e+27)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9e+27)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e+27], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.90000000000000011e27

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 63.7%

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

   if 1.90000000000000011e27 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.5%

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

   4. Taylor expanded in re around 0 69.8%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 78.6%

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

   6. Simplified 69.8%

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

Alternative 9: 32.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.5e-11) 1.0 (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.5e-11) {
		tmp = 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 <= 8.5d-11) then
        tmp = 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 <= 8.5e-11) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.5e-11:
		tmp = 1.0
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.5e-11)
		tmp = 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 <= 8.5e-11)
		tmp = 1.0;
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.5e-11], 1.0, 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 < 8.50000000000000037e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.5%

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

   4. Taylor expanded in re around 0 38.7%

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

   if 8.50000000000000037e-11 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 7.3%

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

      1. distribute-rgt1-in 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in im around 0 11.2%

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

      1. associate-+r+ 11.2%

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

      2. +-commutative 11.2%

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

      3. *-lft-identity 11.2%

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

      4. associate-*r* 11.2%

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

      5. +-commutative 11.2%

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

      6. distribute-rgt-out 11.2%

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

   8. Simplified 11.2%

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

      1. unpow2 11.2%

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

   10. Applied egg-rr 11.2%

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

   11. Taylor expanded in re around inf 11.2%

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

Alternative 10: 40.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	return 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
}

Python

def code(re, im):
	return 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.0%

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

4. Taylor expanded in re around 0 44.4%

   \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 67.3%

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

6. Simplified 44.4%

   \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
7. Add Preprocessing

Alternative 11: 37.3% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 76.0%

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

4. Taylor expanded in re around 0 40.7%

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

   1. *-commutative 63.2%

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

6. Simplified 40.7%

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

Alternative 12: 28.5% 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 im around 0 76.0%

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

4. Taylor expanded in re around 0 30.1%

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

   1. +-commutative 30.1%

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

6. Simplified 30.1%

   \[\leadsto \color{blue}{re + 1} \]
7. Add Preprocessing

Alternative 13: 28.1% 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 im around 0 76.0%

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

4. Taylor expanded in re around 0 29.6%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))