math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 8

Speedup: 1.0×

• 24.0% 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: 97.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0088 \lor \neg \left(re \leq 132000000\right) \land re \leq 1.8 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\cos im \cdot \left(1 + re \cdot \left(0.3333333333333333 + re \cdot 0.05555555555555555\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0088) (and (not (<= re 132000000.0)) (<= re 1.8e+103)))
   (exp re)
   (*
    (+ 1.0 (* re 0.6666666666666666))
    (*
     (cos im)
     (+ 1.0 (* re (+ 0.3333333333333333 (* re 0.05555555555555555))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0088) || (!(re <= 132000000.0) && (re <= 1.8e+103))) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (re * 0.6666666666666666)) * (cos(im) * (1.0 + (re * (0.3333333333333333 + (re * 0.05555555555555555)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0088d0)) .or. (.not. (re <= 132000000.0d0)) .and. (re <= 1.8d+103)) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + (re * 0.6666666666666666d0)) * (cos(im) * (1.0d0 + (re * (0.3333333333333333d0 + (re * 0.05555555555555555d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0088) || (!(re <= 132000000.0) && (re <= 1.8e+103))) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (re * 0.6666666666666666)) * (Math.cos(im) * (1.0 + (re * (0.3333333333333333 + (re * 0.05555555555555555)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0088) or (not (re <= 132000000.0) and (re <= 1.8e+103)):
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (re * 0.6666666666666666)) * (math.cos(im) * (1.0 + (re * (0.3333333333333333 + (re * 0.05555555555555555)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0088) || (!(re <= 132000000.0) && (re <= 1.8e+103)))
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(re * 0.6666666666666666)) * Float64(cos(im) * Float64(1.0 + Float64(re * Float64(0.3333333333333333 + Float64(re * 0.05555555555555555))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0088) || (~((re <= 132000000.0)) && (re <= 1.8e+103)))
		tmp = exp(re);
	else
		tmp = (1.0 + (re * 0.6666666666666666)) * (cos(im) * (1.0 + (re * (0.3333333333333333 + (re * 0.05555555555555555)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0088], And[N[Not[LessEqual[re, 132000000.0]], $MachinePrecision], LessEqual[re, 1.8e+103]]], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(re * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(0.3333333333333333 + N[(re * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00880000000000000053 or 1.32e8 < re < 1.80000000000000008e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.6%

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

   if -0.00880000000000000053 < re < 1.32e8 or 1.80000000000000008e103 < re

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{e^{re} \cdot \cos im} \cdot \sqrt[3]{e^{re} \cdot \cos im}\right) \cdot \sqrt[3]{e^{re} \cdot \cos im}} \]

      2. pow3 99.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re} \cdot \cos im}\right)}^{3}} \]

      3. pow-to-exp 70.8%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{e^{re} \cdot \cos im}\right) \cdot 3}} \]

      4. pow1/3 71.1%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \cos im\right)}^{0.3333333333333333}\right)} \cdot 3} \]

      5. log-pow 71.3%

         \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{re} \cdot \cos im\right)\right)} \cdot 3} \]

      6. log-prod 71.3%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \cos im\right)}\right) \cdot 3} \]

      7. add-log-exp 71.3%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{re} + \log \cos im\right)\right) \cdot 3} \]

   4. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(re + \log \cos im\right)\right) \cdot 3}} \]
   5. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto e^{\color{blue}{3 \cdot \left(0.3333333333333333 \cdot \left(re + \log \cos im\right)\right)}} \]

      2. associate-*r* 71.3%

         \[\leadsto e^{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \left(re + \log \cos im\right)}} \]

      3. metadata-eval 71.3%

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

      4. *-un-lft-identity 71.3%

         \[\leadsto e^{\color{blue}{re + \log \cos im}} \]

      5. prod-exp 71.3%

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

      6. add-exp-log 100.0%

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

      7. add-cube-cbrt 99.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \sqrt[3]{e^{re}}\right)} \cdot \cos im \]

      8. associate-*l* 99.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right)} \]

      9. pow2 99.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2}} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right)} \]

   7. Taylor expanded in re around 0 98.8%

      \[\leadsto \color{blue}{\left(1 + 0.6666666666666666 \cdot re\right)} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]
   8. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \left(1 + \color{blue}{re \cdot 0.6666666666666666}\right) \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]

   9. Simplified 98.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot 0.6666666666666666\right)} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]

   10. Taylor expanded in re around 0 98.3%

       \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\color{blue}{\left(1 + \left(0.05555555555555555 \cdot {re}^{2} + 0.3333333333333333 \cdot re\right)\right)} \cdot \cos im\right) \]
   11. Step-by-step derivation

       1. +-commutative 98.3%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \color{blue}{\left(0.3333333333333333 \cdot re + 0.05555555555555555 \cdot {re}^{2}\right)}\right) \cdot \cos im\right) \]

       2. *-commutative 98.3%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \left(\color{blue}{re \cdot 0.3333333333333333} + 0.05555555555555555 \cdot {re}^{2}\right)\right) \cdot \cos im\right) \]

       3. *-commutative 98.3%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \left(re \cdot 0.3333333333333333 + \color{blue}{{re}^{2} \cdot 0.05555555555555555}\right)\right) \cdot \cos im\right) \]

       4. unpow2 98.3%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \left(re \cdot 0.3333333333333333 + \color{blue}{\left(re \cdot re\right)} \cdot 0.05555555555555555\right)\right) \cdot \cos im\right) \]

       5. associate-*l* 98.3%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \left(re \cdot 0.3333333333333333 + \color{blue}{re \cdot \left(re \cdot 0.05555555555555555\right)}\right)\right) \cdot \cos im\right) \]

       6. distribute-lft-out 98.3%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \color{blue}{re \cdot \left(0.3333333333333333 + re \cdot 0.05555555555555555\right)}\right) \cdot \cos im\right) \]

   12. Simplified 98.3%

       \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\color{blue}{\left(1 + re \cdot \left(0.3333333333333333 + re \cdot 0.05555555555555555\right)\right)} \cdot \cos im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0088 \lor \neg \left(re \leq 132000000\right) \land re \leq 1.8 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\cos im \cdot \left(1 + re \cdot \left(0.3333333333333333 + re \cdot 0.05555555555555555\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.009 \lor \neg \left(re \leq 132000000\right) \land re \leq 6 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\cos im \cdot \left(1 + re \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.009) (and (not (<= re 132000000.0)) (<= re 6e+149)))
   (exp re)
   (*
    (+ 1.0 (* re 0.6666666666666666))
    (* (cos im) (+ 1.0 (* re 0.3333333333333333))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.009) || (!(re <= 132000000.0) && (re <= 6e+149))) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (re * 0.6666666666666666)) * (cos(im) * (1.0 + (re * 0.3333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.009d0)) .or. (.not. (re <= 132000000.0d0)) .and. (re <= 6d+149)) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + (re * 0.6666666666666666d0)) * (cos(im) * (1.0d0 + (re * 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.009) || (!(re <= 132000000.0) && (re <= 6e+149))) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (re * 0.6666666666666666)) * (Math.cos(im) * (1.0 + (re * 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.009) or (not (re <= 132000000.0) and (re <= 6e+149)):
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (re * 0.6666666666666666)) * (math.cos(im) * (1.0 + (re * 0.3333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.009) || (!(re <= 132000000.0) && (re <= 6e+149)))
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(re * 0.6666666666666666)) * Float64(cos(im) * Float64(1.0 + Float64(re * 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.009) || (~((re <= 132000000.0)) && (re <= 6e+149)))
		tmp = exp(re);
	else
		tmp = (1.0 + (re * 0.6666666666666666)) * (cos(im) * (1.0 + (re * 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.009], And[N[Not[LessEqual[re, 132000000.0]], $MachinePrecision], LessEqual[re, 6e+149]]], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(re * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00899999999999999932 or 1.32e8 < re < 6.00000000000000007e149

   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 -0.00899999999999999932 < re < 1.32e8 or 6.00000000000000007e149 < re

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{e^{re} \cdot \cos im} \cdot \sqrt[3]{e^{re} \cdot \cos im}\right) \cdot \sqrt[3]{e^{re} \cdot \cos im}} \]

      2. pow3 99.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re} \cdot \cos im}\right)}^{3}} \]

      3. pow-to-exp 71.4%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{e^{re} \cdot \cos im}\right) \cdot 3}} \]

      4. pow1/3 71.7%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \cos im\right)}^{0.3333333333333333}\right)} \cdot 3} \]

      5. log-pow 71.9%

         \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{re} \cdot \cos im\right)\right)} \cdot 3} \]

      6. log-prod 71.9%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \cos im\right)}\right) \cdot 3} \]

      7. add-log-exp 71.9%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{re} + \log \cos im\right)\right) \cdot 3} \]

   4. Applied egg-rr 71.9%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(re + \log \cos im\right)\right) \cdot 3}} \]
   5. Step-by-step derivation

      1. *-commutative 71.9%

         \[\leadsto e^{\color{blue}{3 \cdot \left(0.3333333333333333 \cdot \left(re + \log \cos im\right)\right)}} \]

      2. associate-*r* 71.9%

         \[\leadsto e^{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \left(re + \log \cos im\right)}} \]

      3. metadata-eval 71.9%

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

      4. *-un-lft-identity 71.9%

         \[\leadsto e^{\color{blue}{re + \log \cos im}} \]

      5. prod-exp 71.9%

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

      6. add-exp-log 100.0%

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

      7. add-cube-cbrt 99.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \sqrt[3]{e^{re}}\right)} \cdot \cos im \]

      8. associate-*l* 99.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right)} \]

      9. pow2 99.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2}} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right)} \]

   7. Taylor expanded in re around 0 98.7%

      \[\leadsto \color{blue}{\left(1 + 0.6666666666666666 \cdot re\right)} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]
   8. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto \left(1 + \color{blue}{re \cdot 0.6666666666666666}\right) \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{\left(1 + re \cdot 0.6666666666666666\right)} \cdot \left(\sqrt[3]{e^{re}} \cdot \cos im\right) \]

   10. Taylor expanded in re around 0 97.1%

       \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\color{blue}{\left(1 + 0.3333333333333333 \cdot re\right)} \cdot \cos im\right) \]
   11. Step-by-step derivation

       1. *-commutative 97.1%

          \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\left(1 + \color{blue}{re \cdot 0.3333333333333333}\right) \cdot \cos im\right) \]

   12. Simplified 97.1%

       \[\leadsto \left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\color{blue}{\left(1 + re \cdot 0.3333333333333333\right)} \cdot \cos im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.009 \lor \neg \left(re \leq 132000000\right) \land re \leq 6 \cdot 10^{+149}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot 0.6666666666666666\right) \cdot \left(\cos im \cdot \left(1 + re \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0155 \lor \neg \left(re \leq 132000000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0155) (not (<= re 132000000.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0155) || !(re <= 132000000.0)) {
		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 <= (-0.0155d0)) .or. (.not. (re <= 132000000.0d0))) 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 <= -0.0155) || !(re <= 132000000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0155) or not (re <= 132000000.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0155) || !(re <= 132000000.0))
		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 <= -0.0155) || ~((re <= 132000000.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0155], N[Not[LessEqual[re, 132000000.0]], $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 < -0.0155 or 1.32e8 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 80.3%

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

   if -0.0155 < re < 1.32e8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.7%

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

      1. distribute-rgt1-in 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0155 \lor \neg \left(re \leq 132000000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{-11} \lor \neg \left(re \leq 2.8 \cdot 10^{-13}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -6.5e-11) (not (<= re 2.8e-13))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -6.5e-11) || !(re <= 2.8e-13)) {
		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 <= (-6.5d-11)) .or. (.not. (re <= 2.8d-13))) 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 <= -6.5e-11) || !(re <= 2.8e-13)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -6.5e-11) or not (re <= 2.8e-13):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -6.5e-11) || !(re <= 2.8e-13))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -6.5e-11) || ~((re <= 2.8e-13)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -6.5e-11], N[Not[LessEqual[re, 2.8e-13]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.49999999999999953e-11 or 2.8000000000000002e-13 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.4%

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

   if -6.49999999999999953e-11 < re < 2.8000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{-11} \lor \neg \left(re \leq 2.8 \cdot 10^{-13}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.4% 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 51.6%

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

4. Final simplification 51.6%

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

Alternative 7: 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 re around 0 52.9%

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

   1. distribute-rgt1-in 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in im around 0 31.5%

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

   1. +-commutative 31.5%

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

8. Simplified 31.5%

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

9. Final simplification 31.5%

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

Alternative 8: 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 52.9%

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

   1. distribute-rgt1-in 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in re around inf 3.7%

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

   1. *-commutative 3.7%

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

8. Simplified 3.7%

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

9. Taylor expanded in im around 0 3.3%

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

10. Final simplification 3.3%

    \[\leadsto re \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))