math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 13

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999998) (not (<= (exp re) 1.0232)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 1.0232)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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) <= 0.9999998d0) .or. (.not. (exp(re) <= 1.0232d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(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) <= 0.9999998) || !(Math.exp(re) <= 1.0232)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999998) or not (math.exp(re) <= 1.0232):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 1.0232))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999998) || ~((exp(re) <= 1.0232)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0232]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999799999999994 or 1.0232000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.7%

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

   if 0.999999799999999994 < (exp.f64 re) < 1.0232000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999998) (not (<= (exp re) 1.0232)))
   (* (exp re) im)
   (/ (sin im) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 1.0232)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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) <= 0.9999998d0) .or. (.not. (exp(re) <= 1.0232d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(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) <= 0.9999998) || !(Math.exp(re) <= 1.0232)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999998) or not (math.exp(re) <= 1.0232):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 1.0232))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999998) || ~((exp(re) <= 1.0232)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0232]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999799999999994 or 1.0232000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.7%

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

   if 0.999999799999999994 < (exp.f64 re) < 1.0232000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

      1. flip-+ 98.9%

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

      2. associate-*l/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. fmm-def 98.9%

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

      5. metadata-eval 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in re around 0 99.0%

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

      1. neg-mul-1 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in im around inf 99.0%

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

       1. mul-1-neg 99.0%

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

       2. sub-neg 99.0%

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

       3. metadata-eval 99.0%

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

       4. distribute-neg-frac2 99.0%

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

       5. +-commutative 99.0%

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

       6. distribute-neg-in 99.0%

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

       7. metadata-eval 99.0%

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

       8. sub-neg 99.0%

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

   13. Simplified 99.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999995 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999999995) (not (<= (exp re) 1.0232)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999999995) || !(exp(re) <= 1.0232)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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) <= 0.999999995d0) .or. (.not. (exp(re) <= 1.0232d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.999999995) || !(Math.exp(re) <= 1.0232)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.999999995) or not (math.exp(re) <= 1.0232):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.999999995) || !(exp(re) <= 1.0232))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.999999995) || ~((exp(re) <= 1.0232)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.999999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0232]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999500000003 or 1.0232000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.8%

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

   if 0.99999999500000003 < (exp.f64 re) < 1.0232000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 97.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999995 \lor \neg \left(e^{re} \leq 1.0232\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{elif}\;re \leq 0.023:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right) + im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-9)
   (/ im (- 1.0 re))
   (if (<= re 0.023)
     (sin im)
     (+
      im
      (* re (+ im (* re (+ (* 0.16666666666666666 (* re im)) (* im 0.5)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-9) {
		tmp = im / (1.0 - re);
	} else if (re <= 0.023) {
		tmp = sin(im);
	} else {
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7.5d-9)) then
        tmp = im / (1.0d0 - re)
    else if (re <= 0.023d0) then
        tmp = sin(im)
    else
        tmp = im + (re * (im + (re * ((0.16666666666666666d0 * (re * im)) + (im * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-9) {
		tmp = im / (1.0 - re);
	} else if (re <= 0.023) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e-9:
		tmp = im / (1.0 - re)
	elif re <= 0.023:
		tmp = math.sin(im)
	else:
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e-9)
		tmp = Float64(im / Float64(1.0 - re));
	elseif (re <= 0.023)
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(re * Float64(im + Float64(re * Float64(Float64(0.16666666666666666 * Float64(re * im)) + Float64(im * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.5e-9)
		tmp = im / (1.0 - re);
	elseif (re <= 0.023)
		tmp = sin(im);
	else
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e-9], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.023], N[Sin[im], $MachinePrecision], N[(im + N[(re * N[(im + N[(re * N[(N[(0.16666666666666666 * N[(re * im), $MachinePrecision]), $MachinePrecision] + N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.49999999999999933e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.8%

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

      1. distribute-rgt1-in 5.8%

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

   5. Simplified 5.8%

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

      1. flip-+ 5.4%

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

      2. associate-*l/ 5.4%

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

      3. metadata-eval 5.4%

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

      4. fmm-def 5.4%

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

      5. metadata-eval 5.4%

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

      6. sub-neg 5.4%

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

      7. metadata-eval 5.4%

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

   7. Applied egg-rr 5.4%

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

   8. Taylor expanded in re around 0 35.7%

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

      1. neg-mul-1 35.7%

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

   10. Simplified 35.7%

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

   11. Taylor expanded in im around 0 34.7%

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

       1. mul-1-neg 34.7%

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

       2. sub-neg 34.7%

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

       3. metadata-eval 34.7%

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

       4. distribute-neg-frac2 34.7%

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

       5. +-commutative 34.7%

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

       6. distribute-neg-in 34.7%

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

       7. metadata-eval 34.7%

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

       8. sub-neg 34.7%

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

   13. Simplified 34.7%

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

   if -7.49999999999999933e-9 < re < 0.023

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 97.4%

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

   if 0.023 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 80.0%

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

   4. Taylor expanded in re around 0 43.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{elif}\;re \leq 0.023:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right) + im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.7% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right) + im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35)
   (/ im (- 1.0 re))
   (+
    im
    (* re (+ im (* re (+ (* 0.16666666666666666 (* re im)) (* im 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35) {
		tmp = im / (1.0 - re);
	} else {
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.35d0)) then
        tmp = im / (1.0d0 - re)
    else
        tmp = im + (re * (im + (re * ((0.16666666666666666d0 * (re * im)) + (im * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.35) {
		tmp = im / (1.0 - re);
	} else {
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.35:
		tmp = im / (1.0 - re)
	else:
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.35)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(im + Float64(re * Float64(im + Float64(re * Float64(Float64(0.16666666666666666 * Float64(re * im)) + Float64(im * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.35)
		tmp = im / (1.0 - re);
	else
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.35], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(im + N[(re * N[(N[(0.16666666666666666 * N[(re * im), $MachinePrecision]), $MachinePrecision] + N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.3500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. flip-+ 2.3%

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

      2. associate-*l/ 2.3%

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

      3. metadata-eval 2.3%

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

      4. fmm-def 2.3%

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

      5. metadata-eval 2.3%

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

      6. sub-neg 2.3%

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

      7. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

   8. Taylor expanded in re around 0 33.7%

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

      1. neg-mul-1 33.7%

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

   10. Simplified 33.7%

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

   11. Taylor expanded in im around 0 32.7%

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

       1. mul-1-neg 32.7%

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

       2. sub-neg 32.7%

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

       3. metadata-eval 32.7%

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

       4. distribute-neg-frac2 32.7%

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

       5. +-commutative 32.7%

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

       6. distribute-neg-in 32.7%

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

       7. metadata-eval 32.7%

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

       8. sub-neg 32.7%

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

   13. Simplified 32.7%

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

   if -1.3500000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.2%

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

   4. Taylor expanded in re around 0 44.6%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right) + im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.2% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(im + Float64(im * 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 <= -1.0)
		tmp = im / (1.0 - re);
	else
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. flip-+ 2.3%

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

      2. associate-*l/ 2.3%

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

      3. metadata-eval 2.3%

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

      4. fmm-def 2.3%

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

      5. metadata-eval 2.3%

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

      6. sub-neg 2.3%

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

      7. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

   8. Taylor expanded in re around 0 33.7%

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

      1. neg-mul-1 33.7%

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

   10. Simplified 33.7%

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

   11. Taylor expanded in im around 0 32.7%

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

       1. mul-1-neg 32.7%

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

       2. sub-neg 32.7%

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

       3. metadata-eval 32.7%

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

       4. distribute-neg-frac2 32.7%

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

       5. +-commutative 32.7%

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

       6. distribute-neg-in 32.7%

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

       7. metadata-eval 32.7%

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

       8. sub-neg 32.7%

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

   13. Simplified 32.7%

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

   if -1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.2%

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

   4. Taylor expanded in re around 0 38.9%

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

   5. Taylor expanded in im around 0 44.5%

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

      1. *-commutative 44.5%

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

   7. Simplified 44.5%

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

4. Final simplification 41.8%

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

Alternative 8: 40.2% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.2e+38) (/ im (- 1.0 re)) (+ im (* re (* im (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.2e+38) {
		tmp = im / (1.0 - re);
	} else {
		tmp = im + (re * (im * (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 <= 4.2d+38) then
        tmp = im / (1.0d0 - re)
    else
        tmp = im + (re * (im * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.2e+38) {
		tmp = im / (1.0 - re);
	} else {
		tmp = im + (re * (im * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.2e+38:
		tmp = im / (1.0 - re)
	else:
		tmp = im + (re * (im * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.2e+38)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(im + Float64(re * Float64(im * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.2e+38)
		tmp = im / (1.0 - re);
	else
		tmp = im + (re * (im * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.2e+38], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.2e38

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 62.9%

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

      1. distribute-rgt1-in 62.9%

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

   5. Simplified 62.9%

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

      1. flip-+ 62.8%

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

      2. associate-*l/ 62.8%

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

      3. metadata-eval 62.8%

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

      4. fmm-def 62.8%

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

      5. metadata-eval 62.8%

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

      6. sub-neg 62.8%

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

      7. metadata-eval 62.8%

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

   7. Applied egg-rr 62.8%

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

   8. Taylor expanded in re around 0 71.5%

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

      1. neg-mul-1 71.5%

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

   10. Simplified 71.5%

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

   11. Taylor expanded in im around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. sub-neg 37.8%

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

       3. metadata-eval 37.8%

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

       4. distribute-neg-frac2 37.8%

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

       5. +-commutative 37.8%

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

       6. distribute-neg-in 37.8%

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

       7. metadata-eval 37.8%

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

       8. sub-neg 37.8%

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

   13. Simplified 37.8%

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

   if 4.2e38 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.0%

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

   4. Taylor expanded in re around 0 36.3%

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

   5. Taylor expanded in re around inf 36.3%

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

      1. *-commutative 36.3%

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

      2. *-commutative 36.3%

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

      3. *-commutative 36.3%

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

      4. associate-*r* 36.3%

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

   7. Simplified 36.3%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.3% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e+23) (* re (/ im re)) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e+23) {
		tmp = re * (im / re);
	} else {
		tmp = re * 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 <= 2.6d+23) then
        tmp = re * (im / re)
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6e+23) {
		tmp = re * (im / re);
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e+23:
		tmp = re * (im / re)
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e+23)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6e+23)
		tmp = re * (im / re);
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e+23], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.59999999999999992e23

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 49.5%

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

      1. distribute-rgt1-in 49.5%

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

   5. Simplified 49.5%

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

   6. Taylor expanded in re around inf 49.4%

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

   7. Taylor expanded in im around 0 31.8%

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

   8. Taylor expanded in re around 0 37.1%

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

   if 2.59999999999999992e23 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 50.9%

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

      1. distribute-rgt1-in 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in re around inf 50.9%

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

   7. Taylor expanded in im around 0 12.1%

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

   8. Taylor expanded in re around inf 12.0%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.0% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.8:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.8) (* re (/ im re)) (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.8) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * 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 <= (-0.8d0)) then
        tmp = re * (im / re)
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.8) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.8:
		tmp = re * (im / re)
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.8)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.8)
		tmp = re * (im / re);
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.8], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

   6. Taylor expanded in re around inf 2.7%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 30.2%

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

   if -0.80000000000000004 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.2%

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

   4. Taylor expanded in re around 0 34.1%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.8:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.5% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.3e+40) (/ im (- 1.0 re)) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+40) {
		tmp = im / (1.0 - re);
	} else {
		tmp = re * 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 <= 3.3d+40) then
        tmp = im / (1.0d0 - re)
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+40) {
		tmp = im / (1.0 - re);
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.3e+40:
		tmp = im / (1.0 - re)
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.3e+40)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.3e+40)
		tmp = im / (1.0 - re);
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.3e+40], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.2999999999999998e40

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 62.9%

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

      1. distribute-rgt1-in 62.9%

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

   5. Simplified 62.9%

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

      1. flip-+ 62.8%

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

      2. associate-*l/ 62.8%

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

      3. metadata-eval 62.8%

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

      4. fmm-def 62.8%

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

      5. metadata-eval 62.8%

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

      6. sub-neg 62.8%

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

      7. metadata-eval 62.8%

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

   7. Applied egg-rr 62.8%

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

   8. Taylor expanded in re around 0 71.5%

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

      1. neg-mul-1 71.5%

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

   10. Simplified 71.5%

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

   11. Taylor expanded in im around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. sub-neg 37.8%

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

       3. metadata-eval 37.8%

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

       4. distribute-neg-frac2 37.8%

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

       5. +-commutative 37.8%

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

       6. distribute-neg-in 37.8%

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

       7. metadata-eval 37.8%

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

       8. sub-neg 37.8%

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

   13. Simplified 37.8%

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

   if 3.2999999999999998e40 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.4%

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

      1. distribute-rgt1-in 4.4%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in re around inf 4.4%

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

   7. Taylor expanded in im around 0 20.1%

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

   8. Taylor expanded in re around inf 20.1%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.6% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.35:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.35) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.35) {
		tmp = im;
	} else {
		tmp = re * 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 <= 1.35d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.35) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.35:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.35)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.35)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.35], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.3500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 62.2%

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

   4. Taylor expanded in re around 0 31.1%

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

   if 1.3500000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in re around inf 4.0%

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

   7. Taylor expanded in im around 0 16.1%

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

   8. Taylor expanded in re around inf 16.1%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.35:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.6% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 67.5%

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

4. Taylor expanded in re around 0 23.0%

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

5. Final simplification 23.0%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))