math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 7

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.995) (not (<= (exp re) 1.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.995) || !(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) <= 0.995d0) .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) <= 0.995) || !(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) <= 0.995) 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) <= 0.995) || !(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) <= 0.995) || ~((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], 0.995], 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) < 0.994999999999999996 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 83.5%

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

   if 0.994999999999999996 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;re \leq -0.00105:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot t\_0}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ re 1.0))))
   (if (<= re -0.00105)
     (exp re)
     (if (<= re 4.9e-30)
       t_0
       (if (<= re 1.7e+154) (exp re) (/ (* re t_0) re))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (re + 1.0);
	double tmp;
	if (re <= -0.00105) {
		tmp = exp(re);
	} else if (re <= 4.9e-30) {
		tmp = t_0;
	} else if (re <= 1.7e+154) {
		tmp = exp(re);
	} else {
		tmp = (re * t_0) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (re + 1.0d0)
    if (re <= (-0.00105d0)) then
        tmp = exp(re)
    else if (re <= 4.9d-30) then
        tmp = t_0
    else if (re <= 1.7d+154) then
        tmp = exp(re)
    else
        tmp = (re * t_0) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (re + 1.0);
	double tmp;
	if (re <= -0.00105) {
		tmp = Math.exp(re);
	} else if (re <= 4.9e-30) {
		tmp = t_0;
	} else if (re <= 1.7e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = (re * t_0) / re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (re + 1.0)
	tmp = 0
	if re <= -0.00105:
		tmp = math.exp(re)
	elif re <= 4.9e-30:
		tmp = t_0
	elif re <= 1.7e+154:
		tmp = math.exp(re)
	else:
		tmp = (re * t_0) / re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (re <= -0.00105)
		tmp = exp(re);
	elseif (re <= 4.9e-30)
		tmp = t_0;
	elseif (re <= 1.7e+154)
		tmp = exp(re);
	else
		tmp = Float64(Float64(re * t_0) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (re + 1.0);
	tmp = 0.0;
	if (re <= -0.00105)
		tmp = exp(re);
	elseif (re <= 4.9e-30)
		tmp = t_0;
	elseif (re <= 1.7e+154)
		tmp = exp(re);
	else
		tmp = (re * t_0) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00105], N[Exp[re], $MachinePrecision], If[LessEqual[re, 4.9e-30], t$95$0, If[LessEqual[re, 1.7e+154], N[Exp[re], $MachinePrecision], N[(N[(re * t$95$0), $MachinePrecision] / re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00104999999999999994 or 4.89999999999999971e-30 < re < 1.69999999999999987e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.9%

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

   if -0.00104999999999999994 < re < 4.89999999999999971e-30

   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%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 1.69999999999999987e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 7.2%

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

      1. distribute-rgt1-in 7.2%

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

      2. *-commutative 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in re around inf 7.2%

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

   7. Taylor expanded in re around 0 7.2%

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

      1. distribute-rgt1-in 7.2%

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

      2. associate-/l* 7.2%

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

      3. +-commutative 7.2%

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

   9. Simplified 7.2%

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

       1. *-commutative 7.2%

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

       2. associate-*r/ 7.2%

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

       3. associate-*l/ 100.0%

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

       4. +-commutative 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00105:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot \left(\cos im \cdot \left(re + 1\right)\right)}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00071 \lor \neg \left(re \leq 4.9 \cdot 10^{-30}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00071) (not (<= re 4.9e-30)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00071) || !(re <= 4.9e-30)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00071d0)) .or. (.not. (re <= 4.9d-30))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00071) || !(re <= 4.9e-30)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00071) or not (re <= 4.9e-30):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00071) || !(re <= 4.9e-30))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00071) || ~((re <= 4.9e-30)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00071], N[Not[LessEqual[re, 4.9e-30]], $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 < -7.10000000000000019e-4 or 4.89999999999999971e-30 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.8%

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

   if -7.10000000000000019e-4 < re < 4.89999999999999971e-30

   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%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00071 \lor \neg \left(re \leq 4.9 \cdot 10^{-30}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return cos(im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[im], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 50.7%

   \[\leadsto \color{blue}{\cos im} \]
4. Add Preprocessing

Alternative 6: 28.4% 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 51.1%

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

   1. distribute-rgt1-in 51.1%

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

   2. *-commutative 51.1%

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

5. Simplified 51.1%

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

6. Taylor expanded in im around 0 27.0%

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

   1. +-commutative 27.0%

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

8. Simplified 27.0%

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

Alternative 7: 28.0% 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 51.1%

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

   1. distribute-rgt1-in 51.1%

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

   2. *-commutative 51.1%

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

5. Simplified 51.1%

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

6. Taylor expanded in re around inf 51.0%

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

7. Taylor expanded in im around 0 26.9%

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

8. Taylor expanded in re around 0 26.6%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))