math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 13

Speedup: 1.0×

• 25.0% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9995) (not (<= (exp re) 2.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.9995d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.9995) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9995) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9995) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99950000000000006 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.2%

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

   if 0.99950000000000006 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.7%

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

      1. distribute-rgt1-in 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9995) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.9995d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.9995) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9995) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9995) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99950000000000006 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.2%

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

   if 0.99950000000000006 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00085:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0086:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00085)
   (exp re)
   (if (<= re 0.0086)
     (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (if (<= re 1.02e+103)
       (exp re)
       (*
        (cos im)
        (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00085) {
		tmp = exp(re);
	} else if (re <= 0.0086) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.02e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00085d0)) then
        tmp = exp(re)
    else if (re <= 0.0086d0) then
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1.02d+103) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00085) {
		tmp = Math.exp(re);
	} else if (re <= 0.0086) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00085:
		tmp = math.exp(re)
	elif re <= 0.0086:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	elif re <= 1.02e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00085)
		tmp = exp(re);
	elseif (re <= 0.0086)
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1.02e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00085)
		tmp = exp(re);
	elseif (re <= 0.0086)
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	elseif (re <= 1.02e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00085], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0086], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.49999999999999953e-4 or 0.0086 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 95.5%

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

   if -8.49999999999999953e-4 < re < 0.0086

   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%

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

   if 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00085:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0086:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00072 \lor \neg \left(re \leq 0.021\right) \land re \leq 6 \cdot 10^{+152}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00072) (and (not (<= re 0.021)) (<= re 6e+152)))
   (exp re)
   (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00072) || (!(re <= 0.021) && (re <= 6e+152))) {
		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 <= (-0.00072d0)) .or. (.not. (re <= 0.021d0)) .and. (re <= 6d+152)) 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 <= -0.00072) || (!(re <= 0.021) && (re <= 6e+152))) {
		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 <= -0.00072) or (not (re <= 0.021) and (re <= 6e+152)):
		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 <= -0.00072) || (!(re <= 0.021) && (re <= 6e+152)))
		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 <= -0.00072) || (~((re <= 0.021)) && (re <= 6e+152)))
		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, -0.00072], And[N[Not[LessEqual[re, 0.021]], $MachinePrecision], LessEqual[re, 6e+152]]], 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 < -7.20000000000000045e-4 or 0.0210000000000000013 < re < 5.99999999999999981e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 95.8%

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

   if -7.20000000000000045e-4 < re < 0.0210000000000000013 or 5.99999999999999981e152 < 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%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00072 \lor \neg \left(re \leq 0.021\right) \land re \leq 6 \cdot 10^{+152}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00055 \lor \neg \left(re \leq 0.00024\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00055) (not (<= re 0.00024)))
   (exp re)
   (* (cos im) (+ -1.0 (+ re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00055) || !(re <= 0.00024)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00055d0)) .or. (.not. (re <= 0.00024d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((-1.0d0) + (re + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00055) || !(re <= 0.00024)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00055) or not (re <= 0.00024):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (-1.0 + (re + 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00055) || !(re <= 0.00024))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(-1.0 + Float64(re + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00055) || ~((re <= 0.00024)))
		tmp = exp(re);
	else
		tmp = cos(im) * (-1.0 + (re + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00055], N[Not[LessEqual[re, 0.00024]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.50000000000000033e-4 or 2.40000000000000006e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.2%

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

   if -5.50000000000000033e-4 < re < 2.40000000000000006e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.7%

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

      1. distribute-rgt1-in 99.7%

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

   5. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-undefine 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

      4. log1p-undefine 99.7%

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

      5. rem-exp-log 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00055 \lor \neg \left(re \leq 0.00024\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.8e-7)
   (cos im)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.8e-7) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.8d-7) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.8e-7) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.8e-7:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.8e-7)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.8e-7)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.8e-7], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.80000000000000015e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 67.5%

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

   if 3.80000000000000015e-7 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.2%

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

   4. Taylor expanded in re around 0 45.1%

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

      1. *-commutative 62.9%

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

   6. Simplified 45.1%

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

Alternative 8: 41.4% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.0%

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

4. Taylor expanded in re around 0 38.4%

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

   1. *-commutative 66.9%

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

6. Simplified 38.4%

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

Alternative 9: 32.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.1 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.1e-35) 1.0 (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.1e-35) {
		tmp = 1.0;
	} 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 <= 8.1d-35) then
        tmp = 1.0d0
    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 <= 8.1e-35) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.1e-35:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.1e-35)
		tmp = 1.0;
	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 <= 8.1e-35)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.1e-35], 1.0, N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.10000000000000029e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.0%

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

   4. Taylor expanded in re around 0 35.2%

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

   if 8.10000000000000029e-35 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 11.4%

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

   4. Taylor expanded in im around 0 19.8%

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

      1. unpow2 19.8%

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

   6. Applied egg-rr 19.8%

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

Alternative 10: 38.3% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.0%

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

4. Taylor expanded in re around 0 35.6%

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

   1. *-commutative 63.7%

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

6. Simplified 35.6%

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

Alternative 11: 37.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.0%

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

4. Taylor expanded in re around 0 35.6%

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

   1. *-commutative 63.7%

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

6. Simplified 35.6%

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

7. Taylor expanded in re around inf 35.6%

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

8. Taylor expanded in re around inf 35.4%

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

Alternative 12: 29.6% 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 52.6%

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

   1. distribute-rgt1-in 52.6%

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

5. Simplified 52.6%

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

6. Taylor expanded in im around 0 28.2%

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

   1. +-commutative 28.2%

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

8. Simplified 28.2%

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

Alternative 13: 29.2% 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 70.0%

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

4. Taylor expanded in re around 0 28.0%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))