math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 15

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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. Add Preprocessing

Alternative 2: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 1.0000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.5) (not (<= (exp re) 1.0000001)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.5) || !(exp(re) <= 1.0000001)) {
		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) <= 0.5d0) .or. (.not. (exp(re) <= 1.0000001d0))) 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) <= 0.5) || !(Math.exp(re) <= 1.0000001)) {
		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) <= 0.5) or not (math.exp(re) <= 1.0000001):
		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) <= 0.5) || !(exp(re) <= 1.0000001))
		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) <= 0.5) || ~((exp(re) <= 1.0000001)))
		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], 0.5], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0000001]], $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) < 0.5 or 1.00000010000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.8%

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

   if 0.5 < (exp.f64 re) < 1.00000010000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 1.0000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 1.0000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.5) (not (<= (exp re) 1.0000001)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.5) || !(exp(re) <= 1.0000001)) {
		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) <= 0.5d0) .or. (.not. (exp(re) <= 1.0000001d0))) 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) <= 0.5) || !(Math.exp(re) <= 1.0000001)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.5) or not (math.exp(re) <= 1.0000001):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.5) || !(exp(re) <= 1.0000001))
		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) <= 0.5) || ~((exp(re) <= 1.0000001)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.5], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0000001]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.5 or 1.00000010000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.8%

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

   if 0.5 < (exp.f64 re) < 1.00000010000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.5 \lor \neg \left(e^{re} \leq 1.0000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.014:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.014)
     t_0
     (if (<= re 1.3e-7)
       (* (sin im) (+ -1.0 (+ re 2.0)))
       (if (<= re 1.05e+103)
         t_0
         (*
          (sin im)
          (+
           1.0
           (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.014) {
		tmp = t_0;
	} else if (re <= 1.3e-7) {
		tmp = sin(im) * (-1.0 + (re + 2.0));
	} else if (re <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = sin(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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.014d0)) then
        tmp = t_0
    else if (re <= 1.3d-7) then
        tmp = sin(im) * ((-1.0d0) + (re + 2.0d0))
    else if (re <= 1.05d+103) then
        tmp = t_0
    else
        tmp = sin(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 t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.014) {
		tmp = t_0;
	} else if (re <= 1.3e-7) {
		tmp = Math.sin(im) * (-1.0 + (re + 2.0));
	} else if (re <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.014:
		tmp = t_0
	elif re <= 1.3e-7:
		tmp = math.sin(im) * (-1.0 + (re + 2.0))
	elif re <= 1.05e+103:
		tmp = t_0
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.014)
		tmp = t_0;
	elseif (re <= 1.3e-7)
		tmp = Float64(sin(im) * Float64(-1.0 + Float64(re + 2.0)));
	elseif (re <= 1.05e+103)
		tmp = t_0;
	else
		tmp = Float64(sin(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)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.014)
		tmp = t_0;
	elseif (re <= 1.3e-7)
		tmp = sin(im) * (-1.0 + (re + 2.0));
	elseif (re <= 1.05e+103)
		tmp = t_0;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.014], t$95$0, If[LessEqual[re, 1.3e-7], N[(N[Sin[im], $MachinePrecision] * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+103], t$95$0, N[(N[Sin[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 3 regimes

2. if re < -0.0140000000000000003 or 1.29999999999999999e-7 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 94.3%

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

   if -0.0140000000000000003 < re < 1.29999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + re\right)\right)} \cdot \sin im \]

      2. expm1-undefine 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + re\right)} - 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.014:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.08:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.08)
     t_0
     (if (<= re 1.3e-7)
       (* (sin im) (+ -1.0 (+ re 2.0)))
       (if (<= re 1.9e+154)
         t_0
         (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.08) {
		tmp = t_0;
	} else if (re <= 1.3e-7) {
		tmp = sin(im) * (-1.0 + (re + 2.0));
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = sin(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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.08d0)) then
        tmp = t_0
    else if (re <= 1.3d-7) then
        tmp = sin(im) * ((-1.0d0) + (re + 2.0d0))
    else if (re <= 1.9d+154) then
        tmp = t_0
    else
        tmp = sin(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 t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.08) {
		tmp = t_0;
	} else if (re <= 1.3e-7) {
		tmp = Math.sin(im) * (-1.0 + (re + 2.0));
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.08:
		tmp = t_0
	elif re <= 1.3e-7:
		tmp = math.sin(im) * (-1.0 + (re + 2.0))
	elif re <= 1.9e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.08)
		tmp = t_0;
	elseif (re <= 1.3e-7)
		tmp = Float64(sin(im) * Float64(-1.0 + Float64(re + 2.0)));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.08)
		tmp = t_0;
	elseif (re <= 1.3e-7)
		tmp = sin(im) * (-1.0 + (re + 2.0));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.08], t$95$0, If[LessEqual[re, 1.3e-7], N[(N[Sin[im], $MachinePrecision] * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0800000000000000017 or 1.29999999999999999e-7 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.6%

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

   if -0.0800000000000000017 < re < 1.29999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + re\right)\right)} \cdot \sin im \]

      2. expm1-undefine 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + re\right)} - 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.08:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0105 \lor \neg \left(re \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0105) (not (<= re 1.3e-7)))
   (* (exp re) im)
   (* (sin im) (+ -1.0 (+ re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0105) || !(re <= 1.3e-7)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0105d0)) .or. (.not. (re <= 1.3d-7))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((-1.0d0) + (re + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0105) || !(re <= 1.3e-7)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0105) or not (re <= 1.3e-7):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (-1.0 + (re + 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0105) || !(re <= 1.3e-7))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(-1.0 + Float64(re + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0105) || ~((re <= 1.3e-7)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (-1.0 + (re + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0105], N[Not[LessEqual[re, 1.3e-7]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0105000000000000007 or 1.29999999999999999e-7 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.8%

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

   if -0.0105000000000000007 < re < 1.29999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + re\right)\right)} \cdot \sin im \]

      2. expm1-undefine 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + re\right)} - 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0105 \lor \neg \left(re \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -215:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{elif}\;re \leq 10^{-7}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))
   (if (<= re -215.0) (* t_0 0.0) (if (<= re 1e-7) (sin im) (* im t_0)))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -215.0) {
		tmp = t_0 * 0.0;
	} else if (re <= 1e-7) {
		tmp = sin(im);
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    if (re <= (-215.0d0)) then
        tmp = t_0 * 0.0d0
    else if (re <= 1d-7) then
        tmp = sin(im)
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -215.0) {
		tmp = t_0 * 0.0;
	} else if (re <= 1e-7) {
		tmp = Math.sin(im);
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	tmp = 0
	if re <= -215.0:
		tmp = t_0 * 0.0
	elif re <= 1e-7:
		tmp = math.sin(im)
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))))
	tmp = 0.0
	if (re <= -215.0)
		tmp = Float64(t_0 * 0.0);
	elseif (re <= 1e-7)
		tmp = sin(im);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	tmp = 0.0;
	if (re <= -215.0)
		tmp = t_0 * 0.0;
	elseif (re <= 1e-7)
		tmp = sin(im);
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -215.0], N[(t$95$0 * 0.0), $MachinePrecision], If[LessEqual[re, 1e-7], N[Sin[im], $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -215

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 1.8%

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

      1. *-commutative 1.8%

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

   5. Simplified 1.8%

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

      1. expm1-log1p-u 1.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 11.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \]

      3. log1p-undefine 11.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \sin im\right)}} - 1\right) \]

      4. rem-exp-log 11.8%

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

   7. Applied egg-rr 11.8%

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

   8. Taylor expanded in im around 0 28.8%

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

   if -215 < re < 9.9999999999999995e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.7%

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

   if 9.9999999999999995e-8 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.0%

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

      1. *-commutative 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in im around 0 61.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -215:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot 0\\ \mathbf{elif}\;re \leq 10^{-7}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))
   (if (<= re -1.6) (* t_0 0.0) (* im t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -1.6) {
		tmp = t_0 * 0.0;
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    if (re <= (-1.6d0)) then
        tmp = t_0 * 0.0d0
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -1.6) {
		tmp = t_0 * 0.0;
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	tmp = 0
	if re <= -1.6:
		tmp = t_0 * 0.0
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))))
	tmp = 0.0
	if (re <= -1.6)
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	tmp = 0.0;
	if (re <= -1.6)
		tmp = t_0 * 0.0;
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.6], N[(t$95$0 * 0.0), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 1.8%

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

      1. *-commutative 1.8%

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

   5. Simplified 1.8%

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

      1. expm1-log1p-u 1.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 11.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \]

      3. log1p-undefine 11.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \sin im\right)}} - 1\right) \]

      4. rem-exp-log 11.8%

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

   7. Applied egg-rr 11.8%

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

   8. Taylor expanded in im around 0 28.8%

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

   if -1.6000000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 91.3%

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

      1. *-commutative 91.3%

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

   5. Simplified 91.3%

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

   6. Taylor expanded in im around 0 51.3%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.9% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

public static double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Python

def code(re, im):
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))

Julia

function code(re, im)
	return Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
end

Wolfram

code[re_, im_] := N[(im * 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. Initial program 100.0%

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

3. Taylor expanded in re around 0 68.2%

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

   1. *-commutative 68.2%

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

5. Simplified 68.2%

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

6. Taylor expanded in im around 0 38.5%

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

7. Final simplification 38.5%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \]
8. Add Preprocessing

Alternative 10: 37.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * 0.5))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
end function

Java

public static double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * 0.5))));
}

Python

def code(re, im):
	return im * (1.0 + (re * (1.0 + (re * 0.5))))

Julia

function code(re, im)
	return Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 65.6%

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

   1. *-commutative 65.6%

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

5. Simplified 65.6%

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

6. Taylor expanded in im around 0 36.5%

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

7. Final simplification 36.5%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 11: 34.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ im (* re (* im (* re 0.5)))))

C

double code(double re, double im) {
	return im + (re * (im * (re * 0.5)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im + (re * (im * (re * 0.5d0)))
end function

Java

public static double code(double re, double im) {
	return im + (re * (im * (re * 0.5)));
}

Python

def code(re, im):
	return im + (re * (im * (re * 0.5)))

Julia

function code(re, im)
	return Float64(im + Float64(re * Float64(im * Float64(re * 0.5))))
end

MATLAB

function tmp = code(re, im)
	tmp = im + (re * (im * (re * 0.5)));
end

Wolfram

code[re_, im_] := N[(im + N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 67.3%

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

4. Taylor expanded in re around 0 33.6%

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

   1. associate-*r* 33.6%

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

   2. *-commutative 33.6%

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

6. Simplified 33.6%

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

7. Taylor expanded in re around inf 33.2%

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

   1. *-commutative 33.2%

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

   2. *-commutative 33.2%

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

   3. *-commutative 33.2%

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

   4. associate-*r* 33.2%

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

9. Simplified 33.2%

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

Alternative 12: 28.2% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+101}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 2.1e+101) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+101) {
		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 (im <= 2.1d+101) 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 (im <= 2.1e+101) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+101:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+101)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+101)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+101], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1e101

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.5%

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

   4. Taylor expanded in re around 0 29.6%

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

   if 2.1e101 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 35.2%

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

   4. Taylor expanded in re around inf 3.9%

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

      1. *-commutative 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in im around 0 7.3%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+101}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.1% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(-1 + \left(re + 2\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ -1.0 (+ re 2.0))))

C

double code(double re, double im) {
	return im * (-1.0 + (re + 2.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((-1.0d0) + (re + 2.0d0))
end function

Java

public static double code(double re, double im) {
	return im * (-1.0 + (re + 2.0));
}

Python

def code(re, im):
	return im * (-1.0 + (re + 2.0))

Julia

function code(re, im)
	return Float64(im * Float64(-1.0 + Float64(re + 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (-1.0 + (re + 2.0));
end

Wolfram

code[re_, im_] := N[(im * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 51.8%

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

   1. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + re\right)\right)} \cdot \sin im \]

   2. expm1-undefine 51.1%

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

5. Applied egg-rr 51.1%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + re\right)} - 1\right)} \cdot \sin im \]
6. Step-by-step derivation

   1. sub-neg 51.1%

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

   2. metadata-eval 51.1%

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

   3. +-commutative 51.1%

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

   4. log1p-undefine 51.1%

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

   5. rem-exp-log 51.8%

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

   6. associate-+r+ 51.8%

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

   7. metadata-eval 51.8%

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

7. Simplified 51.8%

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

8. Taylor expanded in im around 0 27.1%

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

9. Final simplification 27.1%

   \[\leadsto im \cdot \left(-1 + \left(re + 2\right)\right) \]
10. Add Preprocessing

Alternative 14: 30.1% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 67.3%

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

4. Taylor expanded in re around 0 27.1%

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

5. Final simplification 27.1%

   \[\leadsto im + re \cdot im \]
6. Add Preprocessing

Alternative 15: 26.5% 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 67.3%

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

4. Taylor expanded in re around 0 24.7%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))