math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. re = 2.245503414199823e+20
  2. im = 5.990695633970278e-255

• 24.7% 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. Add Preprocessing

Alternative 2: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999995 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999995) (not (<= (exp re) 1.6))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999995) || !(exp(re) <= 1.6)) {
		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) <= 0.9999995d0) .or. (.not. (exp(re) <= 1.6d0))) 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) <= 0.9999995) || !(Math.exp(re) <= 1.6)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999995) or not (math.exp(re) <= 1.6):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999995) || !(exp(re) <= 1.6))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999995) || ~((exp(re) <= 1.6)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.6]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999500000000041 or 1.6000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.5%

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

   if 0.999999500000000041 < (exp.f64 re) < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999995 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999995:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;e^{re} \leq 1.6:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.9999995)
   (+ 1.0 (expm1 re))
   (if (<= (exp re) 1.6) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.9999995) {
		tmp = 1.0 + expm1(re);
	} else if (exp(re) <= 1.6) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.9999995) {
		tmp = 1.0 + Math.expm1(re);
	} else if (Math.exp(re) <= 1.6) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.9999995:
		tmp = 1.0 + math.expm1(re)
	elif math.exp(re) <= 1.6:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.9999995)
		tmp = Float64(1.0 + expm1(re));
	elseif (exp(re) <= 1.6)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.9999995], N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.6], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.999999500000000041

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 97.2%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 97.2%

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

      2. log1p-undefine 97.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \]

      3. add-exp-log 97.2%

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

   5. Applied egg-rr 97.2%

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

   if 0.999999500000000041 < (exp.f64 re) < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.1%

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

   if 1.6000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.5%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.041:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42 \lor \neg \left(re \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.041)
   (+ 1.0 (expm1 re))
   (if (or (<= re 0.42) (not (<= re 1.02e+103)))
     (*
      (cos im)
      (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.041) {
		tmp = 1.0 + expm1(re);
	} else if ((re <= 0.42) || !(re <= 1.02e+103)) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.041) {
		tmp = 1.0 + Math.expm1(re);
	} else if ((re <= 0.42) || !(re <= 1.02e+103)) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.041:
		tmp = 1.0 + math.expm1(re)
	elif (re <= 0.42) or not (re <= 1.02e+103):
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.041)
		tmp = Float64(1.0 + expm1(re));
	elseif ((re <= 0.42) || !(re <= 1.02e+103))
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.041], N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 0.42], N[Not[LessEqual[re, 1.02e+103]], $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], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0410000000000000017

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.6%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.6%

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

      2. log1p-undefine 98.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \]

      3. add-exp-log 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -0.0410000000000000017 < re < 0.419999999999999984 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if 0.419999999999999984 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.0%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.041:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42 \lor \neg \left(re \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.024:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.024)
   (+ 1.0 (expm1 re))
   (if (or (<= re 0.42) (not (<= re 1.9e+154)))
     (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.024) {
		tmp = 1.0 + expm1(re);
	} else if ((re <= 0.42) || !(re <= 1.9e+154)) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.024) {
		tmp = 1.0 + Math.expm1(re);
	} else if ((re <= 0.42) || !(re <= 1.9e+154)) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.024:
		tmp = 1.0 + math.expm1(re)
	elif (re <= 0.42) or not (re <= 1.9e+154):
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.024)
		tmp = Float64(1.0 + expm1(re));
	elseif ((re <= 0.42) || !(re <= 1.9e+154))
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.024], N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 0.42], N[Not[LessEqual[re, 1.9e+154]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.024

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.6%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.6%

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

      2. log1p-undefine 98.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \]

      3. add-exp-log 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -0.024 < re < 0.419999999999999984 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if 0.419999999999999984 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.9%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.024:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00094:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42:\\ \;\;\;\;\frac{\cos im}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00094)
   (+ 1.0 (expm1 re))
   (if (<= re 0.42)
     (/ (cos im) (+ 1.0 (* re (+ (* re 0.5) -1.0))))
     (if (<= re 1.9e+154)
       (exp re)
       (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00094) {
		tmp = 1.0 + expm1(re);
	} else if (re <= 0.42) {
		tmp = cos(im) / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00094) {
		tmp = 1.0 + Math.expm1(re);
	} else if (re <= 0.42) {
		tmp = Math.cos(im) / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else if (re <= 1.9e+154) {
		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 <= -0.00094:
		tmp = 1.0 + math.expm1(re)
	elif re <= 0.42:
		tmp = math.cos(im) / (1.0 + (re * ((re * 0.5) + -1.0)))
	elif re <= 1.9e+154:
		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 <= -0.00094)
		tmp = Float64(1.0 + expm1(re));
	elseif (re <= 0.42)
		tmp = Float64(cos(im) / Float64(1.0 + Float64(re * Float64(Float64(re * 0.5) + -1.0))));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00094], N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.42], N[(N[Cos[im], $MachinePrecision] / N[(1.0 + N[(re * N[(N[(re * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], 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 4 regimes

2. if re < -9.39999999999999972e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.6%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.6%

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

      2. log1p-undefine 98.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \]

      3. add-exp-log 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -9.39999999999999972e-4 < re < 0.419999999999999984

   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. expm1-log1p-u 100.0%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. expm1-undefine 100.0%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - 1\right)}} \cdot \cos im \]

      5. pow-sub 99.9%

         \[\leadsto \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(re\right)}\right)}}{{\left(e^{1}\right)}^{1}}} \cdot \cos im \]

      6. exp-1-e 99.9%

         \[\leadsto \frac{{\color{blue}{e}}^{\left(e^{\mathsf{log1p}\left(re\right)}\right)}}{{\left(e^{1}\right)}^{1}} \cdot \cos im \]

      7. log1p-undefine 99.9%

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

      8. rem-exp-log 99.9%

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

      9. pow1 99.9%

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

      10. exp-1-e 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. clear-num 99.9%

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

      5. pow1 99.9%

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

      6. pow-div 100.0%

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

      7. add-exp-log 100.0%

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

      8. expm1-undefine 100.0%

         \[\leadsto \frac{\cos im}{\frac{1}{{e}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + re\right)\right)\right)}}}} \]

      9. log1p-define 100.0%

         \[\leadsto \frac{\cos im}{\frac{1}{{e}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)\right)}}} \]

      10. expm1-log1p-u 100.0%

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

      11. pow-to-exp 100.0%

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

      12. log-E 100.0%

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

      13. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 99.4%

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

   if 0.419999999999999984 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00094:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42:\\ \;\;\;\;\frac{\cos im}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.8e-5)
   (+ 1.0 (expm1 re))
   (if (<= re 0.42) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.8e-5) {
		tmp = 1.0 + expm1(re);
	} else if (re <= 0.42) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.8e-5) {
		tmp = 1.0 + Math.expm1(re);
	} else if (re <= 0.42) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.8e-5:
		tmp = 1.0 + math.expm1(re)
	elif re <= 0.42:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.8e-5)
		tmp = Float64(1.0 + expm1(re));
	elseif (re <= 0.42)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.8e-5], N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.42], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.8e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.6%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.6%

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

      2. log1p-undefine 98.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \]

      3. add-exp-log 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -5.8e-5 < re < 0.419999999999999984

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

   if 0.419999999999999984 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.5%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;1 + \mathsf{expm1}\left(re\right)\\ \mathbf{elif}\;re \leq 0.42:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.64:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+92}:\\ \;\;\;\;\left(re \cdot \left(1 + \frac{1}{re}\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.64)
   (cos im)
   (if (<= re 1.02e+92)
     (* (* re (+ 1.0 (/ 1.0 re))) (+ 1.0 (* -0.5 (* im im))))
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.64) {
		tmp = cos(im);
	} else if (re <= 1.02e+92) {
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + (-0.5 * (im * 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 <= 0.64d0) then
        tmp = cos(im)
    else if (re <= 1.02d+92) then
        tmp = (re * (1.0d0 + (1.0d0 / re))) * (1.0d0 + ((-0.5d0) * (im * 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 <= 0.64) {
		tmp = Math.cos(im);
	} else if (re <= 1.02e+92) {
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + (-0.5 * (im * 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 <= 0.64:
		tmp = math.cos(im)
	elif re <= 1.02e+92:
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + (-0.5 * (im * 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 <= 0.64)
		tmp = cos(im);
	elseif (re <= 1.02e+92)
		tmp = Float64(Float64(re * Float64(1.0 + Float64(1.0 / re))) * Float64(1.0 + Float64(-0.5 * Float64(im * 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 <= 0.64)
		tmp = cos(im);
	elseif (re <= 1.02e+92)
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + (-0.5 * (im * 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, 0.64], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.02e+92], N[(N[(re * N[(1.0 + N[(1.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if re < 0.640000000000000013

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 71.8%

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

   if 0.640000000000000013 < re < 1.02000000000000003e92

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.3%

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

      1. distribute-rgt1-in 4.3%

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

   5. Simplified 4.3%

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

   6. Taylor expanded in im around 0 23.4%

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

      1. unpow2 23.4%

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

   8. Applied egg-rr 23.4%

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

   9. Taylor expanded in re around inf 23.4%

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

   if 1.02000000000000003e92 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.9%

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

      1. *-commutative 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in im around 0 75.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.64:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+92}:\\ \;\;\;\;\left(re \cdot \left(1 + \frac{1}{re}\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.3e-15) (/ 1.0 (- 1.0 re)) (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.3e-15) {
		tmp = 1.0 / (1.0 - re);
	} 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 <= 1.3d-15) then
        tmp = 1.0d0 / (1.0d0 - re)
    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 <= 1.3e-15) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.3e-15:
		tmp = 1.0 / (1.0 - re)
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.3e-15)
		tmp = Float64(1.0 / Float64(1.0 - re));
	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 <= 1.3e-15)
		tmp = 1.0 / (1.0 - re);
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.3e-15], N[(1.0 / N[(1.0 - re), $MachinePrecision]), $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 < 1.30000000000000002e-15

   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. expm1-log1p-u 72.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. expm1-undefine 72.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - 1\right)}} \cdot \cos im \]

      5. pow-sub 72.5%

         \[\leadsto \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(re\right)}\right)}}{{\left(e^{1}\right)}^{1}}} \cdot \cos im \]

      6. exp-1-e 72.5%

         \[\leadsto \frac{{\color{blue}{e}}^{\left(e^{\mathsf{log1p}\left(re\right)}\right)}}{{\left(e^{1}\right)}^{1}} \cdot \cos im \]

      7. log1p-undefine 72.5%

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

      8. rem-exp-log 99.9%

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

      9. pow1 99.9%

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

      10. exp-1-e 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. clear-num 99.9%

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

      5. pow1 99.9%

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

      6. pow-div 100.0%

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

      7. add-exp-log 72.6%

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

      8. expm1-undefine 72.6%

         \[\leadsto \frac{\cos im}{\frac{1}{{e}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + re\right)\right)\right)}}}} \]

      9. log1p-define 72.6%

         \[\leadsto \frac{\cos im}{\frac{1}{{e}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)\right)}}} \]

      10. expm1-log1p-u 99.9%

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

      11. pow-to-exp 100.0%

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

      12. log-E 100.0%

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

      13. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 73.7%

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

      1. neg-mul-1 73.7%

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

      2. sub-neg 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in im around 0 37.3%

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

   if 1.30000000000000002e-15 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 7.1%

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

      1. distribute-rgt1-in 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in im around 0 18.0%

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

      1. unpow2 18.0%

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

   8. Applied egg-rr 18.0%

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

   9. Taylor expanded in re around inf 18.0%

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

Alternative 10: 41.1% 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 re around 0 71.0%

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

   1. *-commutative 71.0%

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

5. Simplified 71.0%

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

6. Taylor expanded in im around 0 39.8%

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

7. Final simplification 39.8%

   \[\leadsto 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \]
8. Add Preprocessing

Alternative 11: 32.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.3e-15) (/ 1.0 (- 1.0 re)) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.3e-15) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = 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 <= 1.3d-15) then
        tmp = 1.0d0 / (1.0d0 - re)
    else
        tmp = 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 <= 1.3e-15) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.3e-15:
		tmp = 1.0 / (1.0 - re)
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.3e-15)
		tmp = Float64(1.0 / Float64(1.0 - re));
	else
		tmp = 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 <= 1.3e-15)
		tmp = 1.0 / (1.0 - re);
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.3e-15], N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.30000000000000002e-15

   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. expm1-log1p-u 72.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. expm1-undefine 72.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(re\right)} - 1\right)}} \cdot \cos im \]

      5. pow-sub 72.5%

         \[\leadsto \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(re\right)}\right)}}{{\left(e^{1}\right)}^{1}}} \cdot \cos im \]

      6. exp-1-e 72.5%

         \[\leadsto \frac{{\color{blue}{e}}^{\left(e^{\mathsf{log1p}\left(re\right)}\right)}}{{\left(e^{1}\right)}^{1}} \cdot \cos im \]

      7. log1p-undefine 72.5%

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

      8. rem-exp-log 99.9%

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

      9. pow1 99.9%

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

      10. exp-1-e 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. clear-num 99.9%

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

      5. pow1 99.9%

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

      6. pow-div 100.0%

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

      7. add-exp-log 72.6%

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

      8. expm1-undefine 72.6%

         \[\leadsto \frac{\cos im}{\frac{1}{{e}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + re\right)\right)\right)}}}} \]

      9. log1p-define 72.6%

         \[\leadsto \frac{\cos im}{\frac{1}{{e}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)\right)}}} \]

      10. expm1-log1p-u 99.9%

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

      11. pow-to-exp 100.0%

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

      12. log-E 100.0%

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

      13. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 73.7%

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

      1. neg-mul-1 73.7%

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

      2. sub-neg 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in im around 0 37.3%

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

   if 1.30000000000000002e-15 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.9%

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

   4. Taylor expanded in im around 0 14.0%

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

      1. unpow2 18.0%

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

   6. Applied egg-rr 14.0%

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

Alternative 12: 38.1% 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 66.0%

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

4. Taylor expanded in re around 0 35.7%

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

   1. *-commutative 66.1%

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

6. Simplified 35.7%

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

Alternative 13: 29.3% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ -1 + \left(re + 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ -1.0 (+ re 2.0)))

C

double code(double re, double im) {
	return -1.0 + (re + 2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) + (re + 2.0d0)
end function

Java

public static double code(double re, double im) {
	return -1.0 + (re + 2.0);
}

Python

def code(re, im):
	return -1.0 + (re + 2.0)

Julia

function code(re, im)
	return Float64(-1.0 + Float64(re + 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0 + (re + 2.0);
end

Wolfram

code[re_, im_] := N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 54.8%

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

   1. distribute-rgt1-in 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in im around 0 27.6%

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

   1. +-commutative 27.6%

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

   2. expm1-log1p-u 27.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re + 1\right)\right)} \]

   3. expm1-undefine 27.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re + 1\right)} - 1} \]

   4. +-commutative 27.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{1 + re}\right)} - 1 \]

8. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 + re\right)} - 1} \]
9. Step-by-step derivation

   1. sub-neg 27.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 + re\right)} + \left(-1\right)} \]

   2. metadata-eval 27.1%

      \[\leadsto e^{\mathsf{log1p}\left(1 + re\right)} + \color{blue}{-1} \]

   3. +-commutative 27.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(1 + re\right)}} \]

   4. log1p-undefine 27.1%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \left(1 + re\right)\right)}} \]

   5. rem-exp-log 27.6%

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

   6. associate-+r+ 27.6%

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

   7. metadata-eval 27.6%

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

10. Simplified 27.6%

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

11. Final simplification 27.6%

    \[\leadsto -1 + \left(re + 2\right) \]
12. Add Preprocessing

Alternative 14: 29.3% 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 54.8%

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

   1. distribute-rgt1-in 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in im around 0 27.6%

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

7. Final simplification 27.6%

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

Alternative 15: 28.8% 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 66.0%

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

4. Taylor expanded in re around 0 27.2%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))