math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 9

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;re + \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999998) (not (<= (exp re) 1.0232)))
   (exp re)
   (+ re (cos im))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 1.0232)) {
		tmp = exp(re);
	} else {
		tmp = re + 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) <= 0.9999998d0) .or. (.not. (exp(re) <= 1.0232d0))) then
        tmp = exp(re)
    else
        tmp = re + cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.9999998) || !(Math.exp(re) <= 1.0232)) {
		tmp = Math.exp(re);
	} else {
		tmp = re + Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999998) or not (math.exp(re) <= 1.0232):
		tmp = math.exp(re)
	else:
		tmp = re + math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 1.0232))
		tmp = exp(re);
	else
		tmp = Float64(re + cos(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999998) || ~((exp(re) <= 1.0232)))
		tmp = exp(re);
	else
		tmp = re + cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0232]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(re + N[Cos[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999799999999994 or 1.0232000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.4%

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

   if 0.999999799999999994 < (exp.f64 re) < 1.0232000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

   4. Taylor expanded in im around 0 97.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;re + \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999995 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999999995) (not (<= (exp re) 1.0232)))
   (exp re)
   (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999999995) || !(exp(re) <= 1.0232)) {
		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) <= 0.999999995d0) .or. (.not. (exp(re) <= 1.0232d0))) 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) <= 0.999999995) || !(Math.exp(re) <= 1.0232)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.999999995) or not (math.exp(re) <= 1.0232):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.999999995) || !(exp(re) <= 1.0232))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.999999995) || ~((exp(re) <= 1.0232)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.999999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0232]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999500000003 or 1.0232000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.6%

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

   if 0.99999999500000003 < (exp.f64 re) < 1.0232000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999995 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0052 \lor \neg \left(re \leq 0.023\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0052) (and (not (<= re 0.023)) (<= 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 <= -0.0052) || (!(re <= 0.023) && (re <= 1.9e+154))) {
		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 <= (-0.0052d0)) .or. (.not. (re <= 0.023d0)) .and. (re <= 1.9d+154)) 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 <= -0.0052) || (!(re <= 0.023) && (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 <= -0.0052) or (not (re <= 0.023) and (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 <= -0.0052) || (!(re <= 0.023) && (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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0052) || (~((re <= 0.023)) && (re <= 1.9e+154)))
		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, -0.0052], And[N[Not[LessEqual[re, 0.023]], $MachinePrecision], 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 2 regimes

2. if re < -0.0051999999999999998 or 0.023 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.2%

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

   if -0.0051999999999999998 < re < 0.023 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.5%

         \[\leadsto e^{re} \cdot \sqrt[3]{\color{blue}{{\cos im}^{3}}} \]

   4. Applied egg-rr 99.5%

      \[\leadsto e^{re} \cdot \color{blue}{\sqrt[3]{{\cos im}^{3}}} \]

   5. Taylor expanded in re around 0 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-+r+ 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.5%

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

      8. associate-*r* 99.5%

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

      9. distribute-rgt-out 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in re around 0 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0052 \lor \neg \left(re \leq 0.023\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-7} \lor \neg \left(re \leq 0.023\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -3.4e-7) (not (<= re 0.023)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -3.4e-7) || !(re <= 0.023)) {
		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 <= (-3.4d-7)) .or. (.not. (re <= 0.023d0))) 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 <= -3.4e-7) || !(re <= 0.023)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -3.4e-7) or not (re <= 0.023):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -3.4e-7) || !(re <= 0.023))
		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 <= -3.4e-7) || ~((re <= 0.023)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -3.4e-7], N[Not[LessEqual[re, 0.023]], $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 < -3.39999999999999974e-7 or 0.023 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.4%

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

   if -3.39999999999999974e-7 < re < 0.023

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-7} \lor \neg \left(re \leq 0.023\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.023:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.023) (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.023) {
		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 <= 0.023d0) 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 <= 0.023) {
		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 <= 0.023:
		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 <= 0.023)
		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 <= 0.023)
		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, 0.023], 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 < 0.023

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 67.2%

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

   if 0.023 < re

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

         \[\leadsto e^{re} \cdot \sqrt[3]{\color{blue}{{\cos im}^{3}}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{\sqrt[3]{{\cos im}^{3}}} \]

   5. Taylor expanded in re around 0 47.4%

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

      1. distribute-lft-in 47.4%

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

      2. *-commutative 47.4%

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

      3. associate-+r+ 47.4%

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

      4. *-rgt-identity 47.4%

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

      5. distribute-lft-in 47.4%

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

      6. *-commutative 47.4%

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

      7. associate-*r* 47.4%

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

      8. associate-*r* 47.4%

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

      9. distribute-rgt-out 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in re around 0 47.4%

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

   9. Taylor expanded in im around 0 32.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.023:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.6% 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-cbrt-cube 99.7%

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

   2. pow3 99.7%

      \[\leadsto e^{re} \cdot \sqrt[3]{\color{blue}{{\cos im}^{3}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto e^{re} \cdot \color{blue}{\sqrt[3]{{\cos im}^{3}}} \]

5. Taylor expanded in re around 0 62.5%

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

   1. distribute-lft-in 62.5%

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

   2. *-commutative 62.5%

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

   3. associate-+r+ 62.4%

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

   4. *-rgt-identity 62.4%

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

   5. distribute-lft-in 62.4%

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

   6. *-commutative 62.4%

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

   7. associate-*r* 62.4%

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

   8. associate-*r* 62.4%

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

   9. distribute-rgt-out 62.4%

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

7. Simplified 62.4%

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

8. Taylor expanded in re around 0 62.4%

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

9. Taylor expanded in im around 0 34.3%

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

10. Final simplification 34.3%

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

Alternative 8: 28.6% 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 50.0%

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

   1. distribute-rgt1-in 50.0%

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

5. Simplified 50.0%

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

6. Taylor expanded in im around 0 26.1%

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

7. Final simplification 26.1%

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

Alternative 9: 28.3% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 48.6%

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

4. Taylor expanded in im around 0 25.5%

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

Reproduce

herbie shell --seed 2024100 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))