math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 13

Speedup: 1.0×

• 24.6% 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: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999995) (not (<= (exp re) 1.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999995) || !(exp(re) <= 1.0)) {
		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) <= 0.99999995d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.99999995) || !(Math.exp(re) <= 1.0)) {
		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) <= 0.99999995) or not (math.exp(re) <= 1.0):
		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) <= 0.99999995) || !(exp(re) <= 1.0))
		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) <= 0.99999995) || ~((exp(re) <= 1.0)))
		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], 0.99999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $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) < 0.999999949999999971 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.0%

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

   if 0.999999949999999971 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999995) (not (<= (exp re) 1.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999995) || !(exp(re) <= 1.0)) {
		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) <= 0.99999995d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.99999995) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99999995) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99999995) || !(exp(re) <= 1.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.99999995) || ~((exp(re) <= 1.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999949999999971 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.0%

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

   if 0.999999949999999971 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.012 \lor \neg \left(re \leq 1.2 \cdot 10^{-20}\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.012) (and (not (<= re 1.2e-20)) (<= 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 <= -0.012) || (!(re <= 1.2e-20) && (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 <= (-0.012d0)) .or. (.not. (re <= 1.2d-20)) .and. (re <= 1.02d+103)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.012) || (!(re <= 1.2e-20) && (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 <= -0.012) or (not (re <= 1.2e-20) and (re <= 1.02e+103)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.012) || (!(re <= 1.2e-20) && (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 <= -0.012) || (~((re <= 1.2e-20)) && (re <= 1.02e+103)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.012], And[N[Not[LessEqual[re, 1.2e-20]], $MachinePrecision], LessEqual[re, 1.02e+103]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.012 or 1.19999999999999996e-20 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 97.5%

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

   if -0.012 < re < 1.19999999999999996e-20 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.012 \lor \neg \left(re \leq 1.2 \cdot 10^{-20}\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0082 \lor \neg \left(re \leq 1.2 \cdot 10^{-20}\right) \land re \leq 9.2 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0082) (and (not (<= re 1.2e-20)) (<= re 9.2e+149)))
   (exp re)
   (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0082) || (!(re <= 1.2e-20) && (re <= 9.2e+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 <= (-0.0082d0)) .or. (.not. (re <= 1.2d-20)) .and. (re <= 9.2d+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 <= -0.0082) || (!(re <= 1.2e-20) && (re <= 9.2e+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 <= -0.0082) or (not (re <= 1.2e-20) and (re <= 9.2e+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 <= -0.0082) || (!(re <= 1.2e-20) && (re <= 9.2e+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 <= -0.0082) || (~((re <= 1.2e-20)) && (re <= 9.2e+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, -0.0082], And[N[Not[LessEqual[re, 1.2e-20]], $MachinePrecision], LessEqual[re, 9.2e+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 < -0.00820000000000000069 or 1.19999999999999996e-20 < re < 9.1999999999999993e149

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 95.2%

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

   if -0.00820000000000000069 < re < 1.19999999999999996e-20 or 9.1999999999999993e149 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0082 \lor \neg \left(re \leq 1.2 \cdot 10^{-20}\right) \land re \leq 9.2 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{re + 1}}{e}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.5e-8)
   (/ (exp (+ re 1.0)) E)
   (if (<= re 1.2e-20) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.5e-8) {
		tmp = exp((re + 1.0)) / ((double) M_E);
	} else if (re <= 1.2e-20) {
		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 <= -8.5e-8) {
		tmp = Math.exp((re + 1.0)) / Math.E;
	} else if (re <= 1.2e-20) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.5e-8:
		tmp = math.exp((re + 1.0)) / math.e
	elif re <= 1.2e-20:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.5e-8)
		tmp = Float64(exp(Float64(re + 1.0)) / exp(1));
	elseif (re <= 1.2e-20)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.5e-8)
		tmp = exp((re + 1.0)) / 2.71828182845904523536;
	elseif (re <= 1.2e-20)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.5e-8], N[(N[Exp[N[(re + 1.0), $MachinePrecision]], $MachinePrecision] / E), $MachinePrecision], If[LessEqual[re, 1.2e-20], 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 < -8.49999999999999935e-8

   1. Initial program 100.0%

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

      1. expm1-log1p-u 5.0%

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

      2. expm1-undefine 5.0%

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

      3. exp-diff 5.0%

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

      4. log1p-undefine 5.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0 99.8%

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

   if -8.49999999999999935e-8 < re < 1.19999999999999996e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

   if 1.19999999999999996e-20 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{re + 1}}{e}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.2e-20)
   (cos im)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.2e-20) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.2d-20) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.2e-20) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.2e-20:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.2e-20)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.2e-20)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.2e-20], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.19999999999999996e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 69.6%

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

   if 1.19999999999999996e-20 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.6%

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

   4. Taylor expanded in re around 0 54.0%

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

      1. *-commutative 73.5%

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

   6. Simplified 54.0%

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

Alternative 8: 41.3% 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 68.4%

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

4. Taylor expanded in re around 0 40.2%

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

   1. *-commutative 71.1%

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

6. Simplified 40.2%

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

Alternative 9: 30.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{e \cdot \left(re + 1\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.15e+105) (/ (* E (+ re 1.0)) E) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.15e+105) {
		tmp = (((double) M_E) * (re + 1.0)) / ((double) M_E);
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.15e+105) {
		tmp = (Math.E * (re + 1.0)) / Math.E;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.15e+105:
		tmp = (math.e * (re + 1.0)) / math.e
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.15e+105)
		tmp = Float64(Float64(exp(1) * Float64(re + 1.0)) / exp(1));
	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 <= 2.15e+105)
		tmp = (2.71828182845904523536 * (re + 1.0)) / 2.71828182845904523536;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.15e+105], N[(N[(E * N[(re + 1.0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.1500000000000001e105

   1. Initial program 100.0%

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

      1. expm1-log1p-u 73.0%

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

      2. expm1-undefine 73.0%

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

      3. exp-diff 73.0%

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

      4. log1p-undefine 73.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0 67.8%

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

   6. Taylor expanded in re around 0 32.9%

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

      1. exp-1-e 32.9%

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

      2. exp-1-e 32.9%

         \[\leadsto \frac{e + re \cdot \color{blue}{e}}{e} \]

      3. distribute-rgt1-in 32.9%

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

      4. +-commutative 32.9%

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

   8. Simplified 32.9%

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

   if 2.1500000000000001e105 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.1%

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

   4. Taylor expanded in im around 0 15.8%

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

      1. unpow2 15.8%

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

   6. Applied egg-rr 15.8%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{e \cdot \left(re + 1\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.2 \cdot 10^{+105}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.2e+105) (+ re 1.0) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.2e+105) {
		tmp = re + 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 <= 8.2d+105) then
        tmp = re + 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 <= 8.2e+105) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.2e+105:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.2e+105)
		tmp = Float64(re + 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 <= 8.2e+105)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.2e+105], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.2000000000000005e105

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 67.9%

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

   4. Taylor expanded in re around 0 32.9%

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

      1. +-commutative 32.9%

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

   6. Simplified 32.9%

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

   if 8.2000000000000005e105 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.1%

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

   4. Taylor expanded in im around 0 15.8%

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

      1. unpow2 15.8%

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

   6. Applied egg-rr 15.8%

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

Alternative 11: 38.5% 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 68.4%

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

4. Taylor expanded in re around 0 39.0%

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

   1. *-commutative 68.8%

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

6. Simplified 39.0%

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

Alternative 12: 29.2% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.4%

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

4. Taylor expanded in re around 0 28.0%

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

   1. +-commutative 28.0%

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

6. Simplified 28.0%

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

Alternative 13: 28.7% 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 68.4%

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

4. Taylor expanded in re around 0 27.3%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))