math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 7

Speedup: 1.0×

• 25.4% 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. Final simplification 100.0%

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

Alternative 2: 92.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.26 \lor \neg \left(re \leq 11200000000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.26) (not (<= re 11200000000.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.26) || !(re <= 11200000000.0)) {
		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 ((re <= (-0.26d0)) .or. (.not. (re <= 11200000000.0d0))) 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 ((re <= -0.26) || !(re <= 11200000000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.26) or not (re <= 11200000000.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.26) || !(re <= 11200000000.0))
		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 ((re <= -0.26) || ~((re <= 11200000000.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.26], N[Not[LessEqual[re, 11200000000.0]], $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 re < -0.26000000000000001 or 1.12e10 < 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 -0.26000000000000001 < re < 1.12e10

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 96.9%

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

      1. distribute-rgt1-in 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.26 \lor \neg \left(re \leq 11200000000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{-15} \lor \neg \left(re \leq 11200000000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.2e-15) (not (<= re 11200000000.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.2e-15) || !(re <= 11200000000.0)) {
		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 ((re <= (-2.2d-15)) .or. (.not. (re <= 11200000000.0d0))) 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 ((re <= -2.2e-15) || !(re <= 11200000000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.2e-15) or not (re <= 11200000000.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.2e-15) || !(re <= 11200000000.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -2.2e-15) || ~((re <= 11200000000.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.2e-15], N[Not[LessEqual[re, 11200000000.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.19999999999999986e-15 or 1.12e10 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.1%

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

   if -2.19999999999999986e-15 < re < 1.12e10

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 96.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{-15} \lor \neg \left(re \leq 11200000000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.0% 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 53.3%

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

4. Final simplification 53.3%

   \[\leadsto \cos im \]
5. Add Preprocessing

Alternative 5: 28.7% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ \left(re + 2\right) + -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ (+ re 2.0) -1.0))

C

double code(double re, double im) {
	return (re + 2.0) + -1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (re + 2.0d0) + (-1.0d0)
end function

Java

public static double code(double re, double im) {
	return (re + 2.0) + -1.0;
}

Python

def code(re, im):
	return (re + 2.0) + -1.0

Julia

function code(re, im)
	return Float64(Float64(re + 2.0) + -1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = (re + 2.0) + -1.0;
end

Wolfram

code[re_, im_] := N[(N[(re + 2.0), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 93.6%

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

   2. expm1-undefine 93.6%

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

   3. log1p-undefine 93.6%

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

   4. rem-exp-log 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in re around 0 54.2%

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

   1. +-commutative 54.2%

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

   2. *-lft-identity 54.2%

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

   3. distribute-rgt-in 54.2%

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

   4. fma-define 54.2%

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

7. Simplified 54.2%

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

8. Taylor expanded in im around 0 32.6%

   \[\leadsto \color{blue}{\left(2 + re\right)} - 1 \]
9. Step-by-step derivation

   1. +-commutative 32.6%

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

10. Simplified 32.6%

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

11. Final simplification 32.6%

    \[\leadsto \left(re + 2\right) + -1 \]
12. Add Preprocessing

Alternative 6: 28.7% 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 54.3%

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

   1. distribute-rgt1-in 54.3%

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

5. Simplified 54.3%

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

6. Taylor expanded in im around 0 32.6%

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

7. Final simplification 32.6%

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

Alternative 7: 28.2% 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. Step-by-step derivation

   1. expm1-log1p-u 93.6%

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

   2. expm1-undefine 93.6%

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

   3. log1p-undefine 93.6%

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

   4. rem-exp-log 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in re around 0 54.2%

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

   1. +-commutative 54.2%

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

   2. *-lft-identity 54.2%

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

   3. distribute-rgt-in 54.2%

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

   4. fma-define 54.2%

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

7. Simplified 54.2%

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

8. Taylor expanded in im around 0 32.6%

   \[\leadsto \color{blue}{\left(2 + re\right)} - 1 \]
9. Step-by-step derivation

   1. +-commutative 32.6%

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

10. Simplified 32.6%

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

11. Taylor expanded in re around 0 32.2%

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

12. Final simplification 32.2%

    \[\leadsto 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))