math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 9

Speedup: 1.0×

• 24.3% 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: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(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) <= 0.0d0) .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) <= 0.0) || !(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) <= 0.0) 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) <= 0.0) || !(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) <= 0.0) || ~((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], 0.0], 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) < 0.0 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 85.7%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0)))
   (exp re)
   (/ (cos im) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) / (1.0 - re);
	}
	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) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) / (1.0 - re)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 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 85.7%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

      1. flip-+ 98.9%

         \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \cos im \]

      2. associate-*l/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. fma-neg 98.9%

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

      5. metadata-eval 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in re around 0 98.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re + -1} \]
   9. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]

   10. Simplified 98.9%

       \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]

   11. Taylor expanded in im around inf 98.9%

       \[\leadsto \color{blue}{-1 \cdot \frac{\cos im}{re - 1}} \]
   12. Step-by-step derivation

       1. mul-1-neg 98.9%

          \[\leadsto \color{blue}{-\frac{\cos im}{re - 1}} \]

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

          \[\leadsto -\frac{\cos im}{re + \color{blue}{-1}} \]

       4. distribute-neg-frac2 98.9%

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

       5. +-commutative 98.9%

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

       6. distribute-neg-in 98.9%

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

       7. metadata-eval 98.9%

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

       8. sub-neg 98.9%

          \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   13. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{\cos im}{1 - re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999999) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999999) || !(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) <= 0.99999999d0) .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) <= 0.99999999) || !(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) <= 0.99999999) 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) <= 0.99999999) || !(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) <= 0.99999999) || ~((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], 0.99999999], 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) < 0.99999998999999995 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 85.3%

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

   if 0.99999998999999995 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.2e+33) (cos im) (* (+ re 1.0) (+ 1.0 (* (* im im) -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.2e+33) {
		tmp = cos(im);
	} else {
		tmp = (re + 1.0) * (1.0 + ((im * im) * -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 <= 5.2d+33) then
        tmp = cos(im)
    else
        tmp = (re + 1.0d0) * (1.0d0 + ((im * im) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.2e+33) {
		tmp = Math.cos(im);
	} else {
		tmp = (re + 1.0) * (1.0 + ((im * im) * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.2e+33:
		tmp = math.cos(im)
	else:
		tmp = (re + 1.0) * (1.0 + ((im * im) * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.2e+33)
		tmp = cos(im);
	else
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(Float64(im * im) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.2e+33)
		tmp = cos(im);
	else
		tmp = (re + 1.0) * (1.0 + ((im * im) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.2e+33], N[Cos[im], $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.1999999999999995e33

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 66.3%

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

   if 5.1999999999999995e33 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.5%

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

      1. distribute-rgt1-in 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in im around 0 29.7%

      \[\leadsto \color{blue}{1 + \left(re + -0.5 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 29.7%

         \[\leadsto \color{blue}{\left(1 + re\right) + -0.5 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)} \]

      2. associate-*r* 29.7%

         \[\leadsto \left(1 + re\right) + \color{blue}{\left(-0.5 \cdot {im}^{2}\right) \cdot \left(1 + re\right)} \]

      3. distribute-rgt1-in 29.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {im}^{2} + 1\right) \cdot \left(1 + re\right)} \]

      4. *-commutative 29.7%

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot -0.5} + 1\right) \cdot \left(1 + re\right) \]

      5. +-commutative 29.7%

         \[\leadsto \left({im}^{2} \cdot -0.5 + 1\right) \cdot \color{blue}{\left(re + 1\right)} \]

   8. Simplified 29.7%

      \[\leadsto \color{blue}{\left({im}^{2} \cdot -0.5 + 1\right) \cdot \left(re + 1\right)} \]
   9. Step-by-step derivation

      1. unpow2 29.7%

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

   10. Applied egg-rr 29.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8:\\ \;\;\;\;\frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8) (/ 1.0 (- 1.0 re)) (* (+ re 1.0) (+ 1.0 (* (* im im) -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = (re + 1.0) * (1.0 + ((im * im) * -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 <= 5.8d0) then
        tmp = 1.0d0 / (1.0d0 - re)
    else
        tmp = (re + 1.0d0) * (1.0d0 + ((im * im) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = (re + 1.0) * (1.0 + ((im * im) * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8:
		tmp = 1.0 / (1.0 - re)
	else:
		tmp = (re + 1.0) * (1.0 + ((im * im) * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8)
		tmp = Float64(1.0 / Float64(1.0 - re));
	else
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(Float64(im * im) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8)
		tmp = 1.0 / (1.0 - re);
	else
		tmp = (re + 1.0) * (1.0 + ((im * im) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8], N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.79999999999999982

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 68.3%

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

      1. distribute-rgt1-in 68.3%

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

   5. Simplified 68.3%

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

      1. flip-+ 68.3%

         \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \cos im \]

      2. associate-*l/ 68.3%

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

      3. metadata-eval 68.3%

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

      4. fma-neg 68.3%

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

      5. metadata-eval 68.3%

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

      6. sub-neg 68.3%

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

      7. metadata-eval 68.3%

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

   7. Applied egg-rr 68.3%

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

   8. Taylor expanded in re around 0 69.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re + -1} \]
   9. Step-by-step derivation

      1. mul-1-neg 69.7%

         \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]

   10. Simplified 69.7%

       \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]

   11. Taylor expanded in im around 0 39.7%

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

       1. metadata-eval 39.7%

          \[\leadsto \frac{\color{blue}{-1}}{re - 1} \]

       2. sub-neg 39.7%

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

       3. metadata-eval 39.7%

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

       4. distribute-neg-frac 39.7%

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

       5. distribute-neg-frac2 39.7%

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

       6. +-commutative 39.7%

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

       7. distribute-neg-in 39.7%

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

       8. metadata-eval 39.7%

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

       9. sub-neg 39.7%

          \[\leadsto \frac{1}{\color{blue}{1 - re}} \]

   13. Simplified 39.7%

       \[\leadsto \color{blue}{\frac{1}{1 - re}} \]

   if 5.79999999999999982 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.3%

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

      1. distribute-rgt1-in 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in im around 0 27.1%

      \[\leadsto \color{blue}{1 + \left(re + -0.5 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 27.1%

         \[\leadsto \color{blue}{\left(1 + re\right) + -0.5 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)} \]

      2. associate-*r* 27.1%

         \[\leadsto \left(1 + re\right) + \color{blue}{\left(-0.5 \cdot {im}^{2}\right) \cdot \left(1 + re\right)} \]

      3. distribute-rgt1-in 27.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {im}^{2} + 1\right) \cdot \left(1 + re\right)} \]

      4. *-commutative 27.1%

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot -0.5} + 1\right) \cdot \left(1 + re\right) \]

      5. +-commutative 27.1%

         \[\leadsto \left({im}^{2} \cdot -0.5 + 1\right) \cdot \color{blue}{\left(re + 1\right)} \]

   8. Simplified 27.1%

      \[\leadsto \color{blue}{\left({im}^{2} \cdot -0.5 + 1\right) \cdot \left(re + 1\right)} \]
   9. Step-by-step derivation

      1. unpow2 27.1%

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

   10. Applied egg-rr 27.1%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8:\\ \;\;\;\;\frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.1% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (- 1.0 re)))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0 / (1.0 - re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 52.1%

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

   1. distribute-rgt1-in 52.1%

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

5. Simplified 52.1%

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

   1. flip-+ 66.0%

      \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \cos im \]

   2. associate-*l/ 66.0%

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

   3. metadata-eval 66.0%

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

   4. fma-neg 66.0%

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

   5. metadata-eval 66.0%

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

   6. sub-neg 66.0%

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

   7. metadata-eval 66.0%

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

7. Applied egg-rr 66.0%

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

8. Taylor expanded in re around 0 52.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re + -1} \]
9. Step-by-step derivation

   1. mul-1-neg 52.0%

      \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]

10. Simplified 52.0%

    \[\leadsto \frac{\color{blue}{-\cos im}}{re + -1} \]

11. Taylor expanded in im around 0 29.8%

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

    1. metadata-eval 29.8%

       \[\leadsto \frac{\color{blue}{-1}}{re - 1} \]

    2. sub-neg 29.8%

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

    3. metadata-eval 29.8%

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

    4. distribute-neg-frac 29.8%

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

    5. distribute-neg-frac2 29.8%

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

    6. +-commutative 29.8%

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

    7. distribute-neg-in 29.8%

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

    8. metadata-eval 29.8%

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

    9. sub-neg 29.8%

       \[\leadsto \frac{1}{\color{blue}{1 - re}} \]

13. Simplified 29.8%

    \[\leadsto \color{blue}{\frac{1}{1 - re}} \]

14. Final simplification 29.8%

    \[\leadsto \frac{1}{1 - re} \]
15. Add Preprocessing

Alternative 8: 30.0% 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 52.1%

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

   1. distribute-rgt1-in 52.1%

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

5. Simplified 52.1%

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

6. Taylor expanded in im around 0 29.6%

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

   1. +-commutative 29.6%

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

8. Simplified 29.6%

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

9. Final simplification 29.6%

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

Alternative 9: 3.5% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 52.1%

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

   1. distribute-rgt1-in 52.1%

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

5. Simplified 52.1%

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

6. Taylor expanded in re around inf 3.8%

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

   1. *-commutative 3.8%

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

8. Simplified 3.8%

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

9. Taylor expanded in im around 0 3.5%

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

10. Final simplification 3.5%

    \[\leadsto re \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))