math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "1.182957290124455326996126838318650571856e270")
  2. reg1 = (bf "4.089879421937521481312379765917202083692e-282")

• 25.2% 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

\[\begin{array}{l} \\ \frac{\cos im}{\frac{1}{e^{re}}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (cos im) (/ 1.0 (exp re))))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return Math.cos(im) / (1.0 / Math.exp(re));
}

Python

def code(re, im):
	return math.cos(im) / (1.0 / math.exp(re))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 71.5%

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

   2. expm1-undefine 71.5%

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

   3. exp-diff 71.5%

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

   4. log1p-undefine 71.5%

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

   5. rem-exp-log 100.0%

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

   6. exp-1-e 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. e-exp-1 100.0%

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

   6. div-exp 100.0%

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

   7. add-exp-log 71.5%

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

   8. expm1-undefine 71.5%

      \[\leadsto \frac{\cos im}{\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\log \left(1 + re\right)\right)}}}} \]

   9. log1p-define 71.5%

      \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)}}} \]

   10. *-rgt-identity 71.5%

       \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{re \cdot 1}\right)\right)}}} \]

   11. expm1-log1p-u 100.0%

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

   12. *-rgt-identity 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\cos im}{\frac{1}{e^{re}}}} \]
7. Add Preprocessing

Alternative 2: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.02 \lor \neg \left(e^{re} \leq 1.01\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.02) (not (<= (exp re) 1.01))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.02) || !(exp(re) <= 1.01)) {
		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) <= 0.02d0) .or. (.not. (exp(re) <= 1.01d0))) 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) <= 0.02) || !(Math.exp(re) <= 1.01)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.02) or not (math.exp(re) <= 1.01):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.02) || !(exp(re) <= 1.01))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.02) || ~((exp(re) <= 1.01)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.02], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.01]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0200000000000000004 or 1.01000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.8%

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

   if 0.0200000000000000004 < (exp.f64 re) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.02 \lor \neg \left(e^{re} \leq 1.01\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 4: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.065 \lor \neg \left(re \leq 0.0126\right) \land re \leq 7.5 \cdot 10^{+82}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.065) (and (not (<= re 0.0126)) (<= re 7.5e+82)))
   (exp re)
   (*
    (cos im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.065) || (!(re <= 0.0126) && (re <= 7.5e+82))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.065d0)) .or. (.not. (re <= 0.0126d0)) .and. (re <= 7.5d+82)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.065) || (!(re <= 0.0126) && (re <= 7.5e+82))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.065) or (not (re <= 0.0126) and (re <= 7.5e+82)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.065) || (!(re <= 0.0126) && (re <= 7.5e+82)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.065) || (~((re <= 0.0126)) && (re <= 7.5e+82)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.065], And[N[Not[LessEqual[re, 0.0126]], $MachinePrecision], LessEqual[re, 7.5e+82]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.065000000000000002 or 0.0126 < re < 7.4999999999999999e82

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 97.8%

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

   if -0.065000000000000002 < re < 0.0126 or 7.4999999999999999e82 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + \color{blue}{re \cdot 0.16666666666666666}\right)\right)\right) \cdot \cos im \]

   5. Simplified 98.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.065 \lor \neg \left(re \leq 0.0126\right) \land re \leq 7.5 \cdot 10^{+82}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.034 \lor \neg \left(re \leq 0.0068\right) \land re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.034) (and (not (<= re 0.0068)) (<= re 1.85e+154)))
   (exp re)
   (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.034) || (!(re <= 0.0068) && (re <= 1.85e+154))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 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 <= (-0.034d0)) .or. (.not. (re <= 0.0068d0)) .and. (re <= 1.85d+154)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.034) || (!(re <= 0.0068) && (re <= 1.85e+154))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.034) or (not (re <= 0.0068) and (re <= 1.85e+154)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.034) || (!(re <= 0.0068) && (re <= 1.85e+154)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.034) || (~((re <= 0.0068)) && (re <= 1.85e+154)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.034], And[N[Not[LessEqual[re, 0.0068]], $MachinePrecision], LessEqual[re, 1.85e+154]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.034000000000000002 or 0.00679999999999999962 < re < 1.84999999999999997e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.2%

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

   if -0.034000000000000002 < re < 0.00679999999999999962 or 1.84999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.034 \lor \neg \left(re \leq 0.0068\right) \land re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.034 \lor \neg \left(re \leq 0.00185\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.034) (not (<= re 0.00185)))
   (exp re)
   (* (cos im) (+ 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.034) || !(re <= 0.00185)) {
		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 ((re <= (-0.034d0)) .or. (.not. (re <= 0.00185d0))) 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 ((re <= -0.034) || !(re <= 0.00185)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.034) or not (re <= 0.00185):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.034) || !(re <= 0.00185))
		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 ((re <= -0.034) || ~((re <= 0.00185)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.034], N[Not[LessEqual[re, 0.00185]], $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 re < -0.034000000000000002 or 0.0018500000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.8%

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

   if -0.034000000000000002 < re < 0.0018500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.034 \lor \neg \left(re \leq 0.00185\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;re \leq 0.00155:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.07)
   (/ 1.0 (+ 1.0 (* re (+ (* re 0.5) -1.0))))
   (if (<= re 0.00155)
     (cos im)
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.07) {
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else if (re <= 0.00155) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.07d0)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((re * 0.5d0) + (-1.0d0))))
    else if (re <= 0.00155d0) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.07) {
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else if (re <= 0.00155) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.07:
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)))
	elif re <= 0.00155:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.07)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(Float64(re * 0.5) + -1.0))));
	elseif (re <= 0.00155)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.07)
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	elseif (re <= 0.00155)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.07], N[(1.0 / N[(1.0 + N[(re * N[(N[(re * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00155], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.070000000000000007

   1. Initial program 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

      3. exp-diff 0.0%

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

      4. log1p-undefine 0.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. e-exp-1 100.0%

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

      6. div-exp 100.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-undefine 0.0%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\log \left(1 + re\right)\right)}}}} \]

      9. log1p-define 0.0%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)}}} \]

      10. *-rgt-identity 0.0%

          \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{re \cdot 1}\right)\right)}}} \]

      11. expm1-log1p-u 100.0%

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

      12. *-rgt-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 53.7%

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

   8. Taylor expanded in im around 0 53.7%

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

   if -0.070000000000000007 < re < 0.00154999999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.4%

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

   if 0.00154999999999999995 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.6%

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

   4. Taylor expanded in re around 0 49.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 72.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + \color{blue}{re \cdot 0.16666666666666666}\right)\right)\right) \cdot \cos im \]

   6. Simplified 49.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;re \leq 0.00155:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1)
   (/ 1.0 (+ 1.0 (* re (+ (* re 0.5) -1.0))))
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1) {
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.1d0)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((re * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.1) {
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1:
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(Float64(re * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.1)
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1], N[(1.0 / N[(1.0 + N[(re * N[(N[(re * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.1000000000000001

   1. Initial program 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

      3. exp-diff 0.0%

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

      4. log1p-undefine 0.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. e-exp-1 100.0%

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

      6. div-exp 100.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-undefine 0.0%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\log \left(1 + re\right)\right)}}}} \]

      9. log1p-define 0.0%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)}}} \]

      10. *-rgt-identity 0.0%

          \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{re \cdot 1}\right)\right)}}} \]

      11. expm1-log1p-u 100.0%

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

      12. *-rgt-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 53.7%

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

   8. Taylor expanded in im around 0 53.7%

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

   if -1.1000000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.4%

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

   4. Taylor expanded in re around 0 50.4%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 89.6%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + \color{blue}{re \cdot 0.16666666666666666}\right)\right)\right) \cdot \cos im \]

   6. Simplified 50.4%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0001:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0001)
   (/ 1.0 (+ 1.0 (* re (+ (* re 0.5) -1.0))))
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0001) {
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 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 <= (-0.0001d0)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((re * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0001) {
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0001:
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)))
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0001)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(Float64(re * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0001)
		tmp = 1.0 / (1.0 + (re * ((re * 0.5) + -1.0)));
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0001], N[(1.0 / N[(1.0 + N[(re * N[(N[(re * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.00000000000000005e-4

   1. Initial program 100.0%

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

      1. expm1-log1p-u 1.3%

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

      2. expm1-undefine 1.4%

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

      3. exp-diff 1.4%

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

      4. log1p-undefine 1.4%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. e-exp-1 100.0%

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

      6. div-exp 100.0%

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

      7. add-exp-log 1.4%

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

      8. expm1-undefine 1.4%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\log \left(1 + re\right)\right)}}}} \]

      9. log1p-define 1.4%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)}}} \]

      10. *-rgt-identity 1.4%

          \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{re \cdot 1}\right)\right)}}} \]

      11. expm1-log1p-u 100.0%

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

      12. *-rgt-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 53.6%

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

   8. Taylor expanded in im around 0 53.3%

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

   if -1.00000000000000005e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   4. Taylor expanded in re around 0 44.9%

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

      1. *-commutative 83.9%

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

   6. Simplified 44.9%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0001:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(re \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.3% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.95e+164)
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))
   (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.95e+164) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.95d+164) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.95e+164) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.95e+164:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.95e+164)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.95e+164)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.95e+164], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.94999999999999993e164

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.9%

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

   4. Taylor expanded in re around 0 35.7%

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

      1. *-commutative 60.0%

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

   6. Simplified 35.7%

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

   if 1.94999999999999993e164 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 39.5%

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

   4. Taylor expanded in im around 0 29.3%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {im}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 29.3%

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

   6. Applied egg-rr 29.3%

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

Alternative 11: 37.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;1 + re \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.95e+164) (+ 1.0 (* re (* re 0.5))) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.95e+164) {
		tmp = 1.0 + (re * (re * 0.5));
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.95d+164) then
        tmp = 1.0d0 + (re * (re * 0.5d0))
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.95e+164) {
		tmp = 1.0 + (re * (re * 0.5));
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.95e+164:
		tmp = 1.0 + (re * (re * 0.5))
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.95e+164)
		tmp = Float64(1.0 + Float64(re * Float64(re * 0.5)));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.95e+164)
		tmp = 1.0 + (re * (re * 0.5));
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.95e+164], N[(1.0 + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.94999999999999993e164

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.9%

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

   4. Taylor expanded in re around 0 35.7%

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

      1. *-commutative 60.0%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in re around inf 35.7%

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

   8. Taylor expanded in re around inf 35.5%

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

   if 1.94999999999999993e164 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 39.5%

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

   4. Taylor expanded in im around 0 29.3%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {im}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 29.3%

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

   6. Applied egg-rr 29.3%

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

Alternative 12: 33.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 28000:\\ \;\;\;\;\frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 28000.0) (/ 1.0 (- 1.0 re)) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 28000.0) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = 1.0 + (-0.5 * (im * 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 <= 28000.0d0) then
        tmp = 1.0d0 / (1.0d0 - re)
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 28000.0) {
		tmp = 1.0 / (1.0 - re);
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 28000.0:
		tmp = 1.0 / (1.0 - re)
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 28000.0)
		tmp = Float64(1.0 / Float64(1.0 - re));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 28000.0)
		tmp = 1.0 / (1.0 - re);
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 28000.0], N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 28000

   1. Initial program 100.0%

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

      1. expm1-log1p-u 61.6%

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

      2. expm1-undefine 61.6%

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

      3. exp-diff 61.6%

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

      4. log1p-undefine 61.6%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. e-exp-1 100.0%

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

      6. div-exp 100.0%

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

      7. add-exp-log 61.6%

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

      8. expm1-undefine 61.6%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\log \left(1 + re\right)\right)}}}} \]

      9. log1p-define 61.6%

         \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)}}} \]

      10. *-rgt-identity 61.6%

          \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{re \cdot 1}\right)\right)}}} \]

      11. expm1-log1p-u 100.0%

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

      12. *-rgt-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 62.3%

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

      1. neg-mul-1 62.3%

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

      2. unsub-neg 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in im around 0 33.4%

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

   if 28000 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.1%

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

   4. Taylor expanded in im around 0 21.9%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {im}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 21.9%

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

   6. Applied egg-rr 21.9%

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

Alternative 13: 29.6% 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. Step-by-step derivation

   1. expm1-log1p-u 71.5%

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

   2. expm1-undefine 71.5%

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

   3. exp-diff 71.5%

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

   4. log1p-undefine 71.5%

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

   5. rem-exp-log 100.0%

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

   6. exp-1-e 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. e-exp-1 100.0%

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

   6. div-exp 100.0%

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

   7. add-exp-log 71.5%

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

   8. expm1-undefine 71.5%

      \[\leadsto \frac{\cos im}{\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\log \left(1 + re\right)\right)}}}} \]

   9. log1p-define 71.5%

      \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(re\right)}\right)}}} \]

   10. *-rgt-identity 71.5%

       \[\leadsto \frac{\cos im}{\frac{1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{re \cdot 1}\right)\right)}}} \]

   11. expm1-log1p-u 100.0%

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

   12. *-rgt-identity 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in re around 0 46.5%

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

   1. neg-mul-1 46.5%

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

   2. unsub-neg 46.5%

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

9. Simplified 46.5%

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

10. Taylor expanded in im around 0 25.1%

    \[\leadsto \color{blue}{\frac{1}{1 - re}} \]
11. Add Preprocessing

Alternative 14: 29.5% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.3%

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

4. Taylor expanded in re around 0 24.7%

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

   1. +-commutative 24.7%

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

6. Simplified 24.7%

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

7. Final simplification 24.7%

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

Alternative 15: 28.9% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.3%

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

4. Taylor expanded in re around 0 24.6%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))