math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 8

Speedup: 1.0×

• 25.6% 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: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.8 \lor \neg \left(e^{re} \leq 1.5\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.8) (not (<= (exp re) 1.5))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.8) || !(exp(re) <= 1.5)) {
		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) <= 0.8d0) .or. (.not. (exp(re) <= 1.5d0))) 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) <= 0.8) || !(Math.exp(re) <= 1.5)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.8) or not (math.exp(re) <= 1.5):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.8) || !(exp(re) <= 1.5))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.8) || ~((exp(re) <= 1.5)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.8], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.5]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.80000000000000004 or 1.5 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.2%

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

   if 0.80000000000000004 < (exp.f64 re) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.8 \lor \neg \left(e^{re} \leq 1.5\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.065 \lor \neg \left(re \leq 0.025\right) \land re \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.065) (and (not (<= re 0.025)) (<= re 7.8e+149)))
   (exp re)
   (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.065) || (!(re <= 0.025) && (re <= 7.8e+149))) {
		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 <= (-0.065d0)) .or. (.not. (re <= 0.025d0)) .and. (re <= 7.8d+149)) 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 <= -0.065) || (!(re <= 0.025) && (re <= 7.8e+149))) {
		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 <= -0.065) or (not (re <= 0.025) and (re <= 7.8e+149)):
		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 <= -0.065) || (!(re <= 0.025) && (re <= 7.8e+149)))
		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 <= -0.065) || (~((re <= 0.025)) && (re <= 7.8e+149)))
		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, -0.065], And[N[Not[LessEqual[re, 0.025]], $MachinePrecision], LessEqual[re, 7.8e+149]]], 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 < -0.065000000000000002 or 0.025000000000000001 < re < 7.7999999999999998e149

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.3%

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

   if -0.065000000000000002 < re < 0.025000000000000001 or 7.7999999999999998e149 < re

   1. Initial program 100.0%

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

      1. add-exp-log 70.1%

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

      2. prod-exp 70.1%

         \[\leadsto \color{blue}{e^{re + \log \cos im}} \]

   4. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{e^{re + \log \cos im}} \]

   5. Taylor expanded in re around 0 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. associate-+r+ 98.5%

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

      3. distribute-rgt1-in 98.5%

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

      4. +-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. associate-*r* 98.5%

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

      7. distribute-rgt-out 98.5%

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

      8. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in re around 0 98.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.065 \lor \neg \left(re \leq 0.025\right) \land re \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00188 \lor \neg \left(re \leq 0.0105\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00188) (not (<= re 0.0105)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00188) || !(re <= 0.0105)) {
		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 ((re <= (-0.00188d0)) .or. (.not. (re <= 0.0105d0))) 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 ((re <= -0.00188) || !(re <= 0.0105)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00188) or not (re <= 0.0105):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00188) || !(re <= 0.0105))
		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 ((re <= -0.00188) || ~((re <= 0.0105)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00188], N[Not[LessEqual[re, 0.0105]], $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 re < -0.00187999999999999994 or 0.0105000000000000007 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.2%

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

   if -0.00187999999999999994 < re < 0.0105000000000000007

   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%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00188 \lor \neg \left(re \leq 0.0105\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.6e-7) (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.6e-7) {
		tmp = cos(im);
	} else {
		tmp = 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 <= 3.6d-7) then
        tmp = cos(im)
    else
        tmp = 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 <= 3.6e-7) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.6e-7:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.6e-7)
		tmp = cos(im);
	else
		tmp = 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 <= 3.6e-7)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.6e-7], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.59999999999999994e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 64.6%

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

   if 3.59999999999999994e-7 < re

   1. Initial program 100.0%

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

      1. add-exp-log 74.2%

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

      2. prod-exp 74.2%

         \[\leadsto \color{blue}{e^{re + \log \cos im}} \]

   4. Applied egg-rr 74.2%

      \[\leadsto \color{blue}{e^{re + \log \cos im}} \]

   5. Taylor expanded in re around 0 57.5%

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

      1. distribute-lft-in 57.5%

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

      2. associate-+r+ 57.5%

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

      3. distribute-rgt1-in 57.5%

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

      4. +-commutative 57.5%

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

      5. associate-*r* 57.5%

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

      6. associate-*r* 57.5%

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

      7. distribute-rgt-out 57.5%

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

      8. +-commutative 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in re around 0 57.5%

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

   9. Taylor expanded in im around 0 41.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.1% 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. Step-by-step derivation

   1. add-exp-log 71.8%

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

   2. prod-exp 71.8%

      \[\leadsto \color{blue}{e^{re + \log \cos im}} \]

4. Applied egg-rr 71.8%

   \[\leadsto \color{blue}{e^{re + \log \cos im}} \]

5. Taylor expanded in re around 0 63.3%

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

   1. distribute-lft-in 63.3%

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

   2. associate-+r+ 63.3%

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

   3. distribute-rgt1-in 63.3%

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

   4. +-commutative 63.3%

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

   5. associate-*r* 63.3%

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

   6. associate-*r* 63.3%

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

   7. distribute-rgt-out 63.3%

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

   8. +-commutative 63.3%

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

7. Simplified 63.3%

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

8. Taylor expanded in re around 0 63.3%

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

9. Taylor expanded in im around 0 36.8%

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

10. Final simplification 36.8%

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

Alternative 7: 28.0% 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 49.7%

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

   1. distribute-rgt1-in 49.7%

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

5. Simplified 49.7%

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

6. Taylor expanded in im around 0 27.2%

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

   1. +-commutative 27.2%

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

8. Simplified 27.2%

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

Alternative 8: 3.5% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 49.7%

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

   1. distribute-rgt1-in 49.7%

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

5. Simplified 49.7%

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

6. Taylor expanded in re around inf 3.6%

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

   1. *-commutative 3.6%

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

8. Simplified 3.6%

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

9. Taylor expanded in im around 0 3.5%

   \[\leadsto \color{blue}{re} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))