math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 9

Speedup: 1.0×

• 24.9% 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. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]

Alternative 2: 92.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 / (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) <= 1.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 / (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) <= 1.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 / (1.0 - re));
	}
	return tmp;
}

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 / 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) <= 1.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 / (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], 1.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 89.8%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

      1. flip-+ 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/l* 99.4%

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

      5. clear-num 99.4%

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

      6. flip-+ 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in re around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. sub-neg 99.4%

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

   9. Simplified 99.4%

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

       1. div-inv 99.4%

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

       2. *-commutative 99.4%

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

   11. Applied egg-rr 99.4%

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

4. Final simplification 94.6%

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

Alternative 3: 92.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(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) <= 0.0d0) .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) <= 0.0) || !(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) <= 0.0) 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) <= 0.0) || !(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) <= 0.0) || ~((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], 0.0], 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) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 89.8%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 94.6%

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

Alternative 4: 92.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0)) {
		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) <= 1.0d0))) 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) <= 1.0)) {
		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) <= 1.0):
		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) <= 1.0))
		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) <= 1.0)))
		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], 1.0]], $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 1 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 89.8%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

      1. flip-+ 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/l* 99.4%

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

      5. clear-num 99.4%

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

      6. flip-+ 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in re around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. sub-neg 99.4%

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

   9. Simplified 99.4%

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

4. Final simplification 94.6%

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

Alternative 5: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999999) (not (<= (exp re) 1.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999999) || !(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) <= 0.999999d0) .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) <= 0.999999) || !(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) <= 0.999999) 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) <= 0.999999) || !(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) <= 0.999999) || ~((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], 0.999999], 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) < 0.999998999999999971 or 1 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 89.2%

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

   if 0.999998999999999971 < (exp.f64 re) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]

Alternative 6: 49.9% 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. Taylor expanded in re around 0 51.8%

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

3. Final simplification 51.8%

   \[\leadsto \cos im \]

Alternative 7: 28.7% 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. Taylor expanded in re around 0 52.3%

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

   1. distribute-rgt1-in 52.3%

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

4. Simplified 52.3%

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

   1. flip-+ 64.3%

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

   2. associate-*l/ 64.3%

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

   3. *-commutative 64.3%

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

   4. associate-/l* 64.3%

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

   5. clear-num 64.3%

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

   6. flip-+ 52.3%

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in re around 0 52.2%

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

   1. mul-1-neg 52.2%

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

   2. sub-neg 52.2%

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

9. Simplified 52.2%

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

10. Taylor expanded in im around 0 30.6%

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

11. Final simplification 30.6%

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

Alternative 8: 28.6% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0 52.3%

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

   1. distribute-rgt1-in 52.3%

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

4. Simplified 52.3%

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

5. Taylor expanded in im around 0 30.3%

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

   1. +-commutative 30.3%

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

7. Simplified 30.3%

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

8. Final simplification 30.3%

   \[\leadsto re + 1 \]

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. Taylor expanded in re around 0 52.3%

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

   1. distribute-rgt1-in 52.3%

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

4. Simplified 52.3%

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

5. Taylor expanded in re around inf 3.6%

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

   1. *-commutative 3.6%

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

7. Simplified 3.6%

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

8. Taylor expanded in im around 0 3.4%

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

9. Final simplification 3.4%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023305 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))