math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 14

Speedup: 1.0×

• 23.4% 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: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.0)))
   (* (exp re) im)
   (* (sin im) (+ -1.0 (+ re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0)) {
		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 ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 1.0d0))) 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 ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 1.0)) {
		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 (math.exp(re) <= 0.0) or not (math.exp(re) <= 1.0):
		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 ((exp(re) <= 0.0) || !(exp(re) <= 1.0))
		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 ((exp(re) <= 0.0) || ~((exp(re) <= 1.0)))
		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[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $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 (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.7%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

      1. distribute-rgt1-in 99.3%

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

   5. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-undefine 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

      4. log1p-undefine 99.3%

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

      5. rem-exp-log 99.3%

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

      6. +-commutative 99.3%

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

      7. associate-+r+ 99.3%

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

      8. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0)) {
		tmp = exp(re) * 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) <= 0.0d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.0) || !(Math.exp(re) <= 1.0)) {
		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) <= 0.0) or not (math.exp(re) <= 1.0):
		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) <= 0.0) || !(exp(re) <= 1.0))
		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) <= 0.0) || ~((exp(re) <= 1.0)))
		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], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $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) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.7%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

      1. distribute-rgt1-in 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 92.2%

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

Alternative 4: 92.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0)) {
		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) <= 0.0d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.0) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.7%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.6%

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

4. Final simplification 91.9%

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

Alternative 5: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.04 \lor \neg \left(re \leq 1.3 \cdot 10^{-8}\right) \land re \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.04) (and (not (<= re 1.3e-8)) (<= re 8.2e+99)))
   (* (exp re) im)
   (*
    (sin im)
    (+ (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.04) || (!(re <= 1.3e-8) && (re <= 8.2e+99))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.04d0)) .or. (.not. (re <= 1.3d-8)) .and. (re <= 8.2d+99)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.04) || (!(re <= 1.3e-8) && (re <= 8.2e+99))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.04) or (not (re <= 1.3e-8) and (re <= 8.2e+99)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.04) || (!(re <= 1.3e-8) && (re <= 8.2e+99)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.04) || (~((re <= 1.3e-8)) && (re <= 8.2e+99)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.04], And[N[Not[LessEqual[re, 1.3e-8]], $MachinePrecision], LessEqual[re, 8.2e+99]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0400000000000000008 or 1.3000000000000001e-8 < re < 8.19999999999999959e99

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.1%

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

   if -0.0400000000000000008 < re < 1.3000000000000001e-8 or 8.19999999999999959e99 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.04 \lor \neg \left(re \leq 1.3 \cdot 10^{-8}\right) \land re \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0105 \lor \neg \left(re \leq 1.3 \cdot 10^{-8}\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0105) (and (not (<= re 1.3e-8)) (<= re 1.9e+154)))
   (* (exp re) im)
   (* (sin im) (+ (* re (+ (* re 0.5) 1.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0105) || (!(re <= 1.3e-8) && (re <= 1.9e+154))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0105d0)) .or. (.not. (re <= 1.3d-8)) .and. (re <= 1.9d+154)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re * ((re * 0.5d0) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0105) || (!(re <= 1.3e-8) && (re <= 1.9e+154))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0105) or (not (re <= 1.3e-8) and (re <= 1.9e+154)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re * ((re * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0105) || (!(re <= 1.3e-8) && (re <= 1.9e+154)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re * Float64(Float64(re * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0105) || (~((re <= 1.3e-8)) && (re <= 1.9e+154)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re * ((re * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0105], And[N[Not[LessEqual[re, 1.3e-8]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * N[(N[(re * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0105000000000000007 or 1.3000000000000001e-8 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.0%

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

   if -0.0105000000000000007 < re < 1.3000000000000001e-8 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0105 \lor \neg \left(re \leq 1.3 \cdot 10^{-8}\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.018 \lor \neg \left(re \leq 1.3 \cdot 10^{-8}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(-1 + \left(re - -3\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.018) (not (<= re 1.3e-8)))
   (* (exp re) im)
   (* (sin im) (+ -1.0 (+ -1.0 (- re -3.0))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.018) || !(re <= 1.3e-8)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (-1.0 + (-1.0 + (re - -3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.018d0)) .or. (.not. (re <= 1.3d-8))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((-1.0d0) + ((-1.0d0) + (re - (-3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.018) || !(re <= 1.3e-8)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (-1.0 + (-1.0 + (re - -3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.018) or not (re <= 1.3e-8):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (-1.0 + (-1.0 + (re - -3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.018) || !(re <= 1.3e-8))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(-1.0 + Float64(-1.0 + Float64(re - -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.018) || ~((re <= 1.3e-8)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (-1.0 + (-1.0 + (re - -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.018], N[Not[LessEqual[re, 1.3e-8]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(-1.0 + N[(-1.0 + N[(re - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0179999999999999986 or 1.3000000000000001e-8 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.7%

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

   if -0.0179999999999999986 < re < 1.3000000000000001e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

      1. distribute-rgt1-in 99.3%

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

   5. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-undefine 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

      4. log1p-undefine 99.3%

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

      5. rem-exp-log 99.3%

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

      6. +-commutative 99.3%

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

      7. associate-+r+ 99.3%

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

      8. metadata-eval 99.3%

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

   9. Simplified 99.3%

      \[\leadsto \color{blue}{\left(-1 + \left(2 + re\right)\right)} \cdot \sin im \]
   10. Step-by-step derivation

       1. expm1-log1p-u 96.2%

          \[\leadsto \left(-1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 + re\right)\right)}\right) \cdot \sin im \]

       2. expm1-undefine 96.2%

          \[\leadsto \left(-1 + \color{blue}{\left(e^{\mathsf{log1p}\left(2 + re\right)} - 1\right)}\right) \cdot \sin im \]

   11. Applied egg-rr 96.2%

       \[\leadsto \left(-1 + \color{blue}{\left(e^{\mathsf{log1p}\left(2 + re\right)} - 1\right)}\right) \cdot \sin im \]
   12. Step-by-step derivation

       1. sub-neg 96.2%

          \[\leadsto \left(-1 + \color{blue}{\left(e^{\mathsf{log1p}\left(2 + re\right)} + \left(-1\right)\right)}\right) \cdot \sin im \]

       2. log1p-undefine 96.2%

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

       3. rem-exp-log 99.3%

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

       4. +-commutative 99.3%

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

       5. metadata-eval 99.3%

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

       6. sub-neg 99.3%

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

       7. metadata-eval 99.3%

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

       8. +-commutative 99.3%

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

       9. sub-neg 99.3%

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

       10. metadata-eval 99.3%

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

       11. +-commutative 99.3%

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

       12. associate-+r+ 99.3%

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

       13. metadata-eval 99.3%

           \[\leadsto \left(-1 + \left(-1 + \left(\color{blue}{3} + re\right)\right)\right) \cdot \sin im \]

       14. metadata-eval 99.3%

           \[\leadsto \left(-1 + \left(-1 + \left(\color{blue}{\left(0 - -3\right)} + re\right)\right)\right) \cdot \sin im \]

       15. associate--r- 99.3%

           \[\leadsto \left(-1 + \left(-1 + \color{blue}{\left(0 - \left(-3 - re\right)\right)}\right)\right) \cdot \sin im \]

       16. neg-sub0 99.3%

           \[\leadsto \left(-1 + \left(-1 + \color{blue}{\left(-\left(-3 - re\right)\right)}\right)\right) \cdot \sin im \]

       17. unsub-neg 99.3%

           \[\leadsto \left(-1 + \color{blue}{\left(-1 - \left(-3 - re\right)\right)}\right) \cdot \sin im \]

   13. Simplified 99.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.018 \lor \neg \left(re \leq 1.3 \cdot 10^{-8}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(-1 + \left(re - -3\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.75e-9)
   (sin im)
   (* im (+ (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.75e-9) {
		tmp = sin(im);
	} else {
		tmp = im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.75d-9) then
        tmp = sin(im)
    else
        tmp = im * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.75e-9) {
		tmp = Math.sin(im);
	} else {
		tmp = im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.75e-9:
		tmp = math.sin(im)
	else:
		tmp = im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.75e-9)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.75e-9)
		tmp = sin(im);
	else
		tmp = im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.75e-9], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re * N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7499999999999998e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 70.4%

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

   if 2.7499999999999998e-9 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.1%

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

   4. Taylor expanded in re around 0 55.8%

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

      1. *-commutative 68.5%

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

   6. Simplified 55.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{+132}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.6e+132)
   (* im (+ (* -0.16666666666666666 (* im im)) 1.0))
   (* im (+ (* re (+ (* re 0.5) 1.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.6e+132) {
		tmp = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	} else {
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.6d+132) then
        tmp = im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)
    else
        tmp = im * ((re * ((re * 0.5d0) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.6e+132) {
		tmp = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	} else {
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.6e+132:
		tmp = im * ((-0.16666666666666666 * (im * im)) + 1.0)
	else:
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.6e+132)
		tmp = Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(im * Float64(Float64(re * Float64(Float64(re * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.6e+132)
		tmp = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	else
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.6e+132], N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * N[(N[(re * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.5999999999999998e132

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 61.5%

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

   4. Taylor expanded in im around 0 34.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 34.4%

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

   6. Applied egg-rr 34.4%

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

   if 5.5999999999999998e132 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.9%

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

   4. Taylor expanded in re around 0 71.4%

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

      1. *-commutative 85.3%

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

   6. Simplified 71.4%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{+132}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.9% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 66.1%

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

4. Taylor expanded in re around 0 40.3%

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

   1. *-commutative 70.1%

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

6. Simplified 40.3%

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

7. Final simplification 40.3%

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

Alternative 11: 30.0% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 52.8%

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

4. Taylor expanded in im around 0 31.5%

   \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
5. Step-by-step derivation

   1. unpow2 31.5%

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

6. Applied egg-rr 31.5%

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

7. Final simplification 31.5%

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

Alternative 12: 28.3% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 5.2e+20) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+20) {
		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 <= 5.2d+20) 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 <= 5.2e+20) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.2e+20:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2e+20)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.2e+20)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2e+20], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.2e20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 51.5%

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

   4. Taylor expanded in im around 0 33.8%

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

   if 5.2e20 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 57.8%

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

      1. distribute-rgt1-in 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in re around inf 3.6%

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

      1. *-commutative 3.6%

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

   8. Simplified 3.6%

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

   9. Taylor expanded in im around 0 8.4%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.9% 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 66.1%

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

4. Taylor expanded in re around 0 29.2%

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

5. Final simplification 29.2%

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

Alternative 14: 26.8% 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 re around 0 52.8%

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

4. Taylor expanded in im around 0 26.8%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))