math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 9

Speedup: 1.5×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(2 \cdot \cosh im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (sin re)) (* 2.0 (cosh im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (2.0 * cosh(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (2.0d0 * cosh(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (2.0 * Math.cosh(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (2.0 * math.cosh(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(2.0 * cosh(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (2.0 * cosh(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. add-log-exp 76.8%

      \[\leadsto \color{blue}{\log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   2. *-un-lft-identity 76.8%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   3. log-prod 76.8%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   4. metadata-eval 76.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   6. +-commutative 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

   7. sub0-neg 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{\color{blue}{-im}}\right) \]

   8. cosh-undef 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \left(0.5 \cdot \sin re\right) \cdot \left(2 \cdot \cosh im\right)} \]

4. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 \cdot \cosh im\right) \]

Alternative 2: 87.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{-5} \lor \neg \left(im \leq 6.2 \cdot 10^{-6}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -9e-5) (not (<= im 6.2e-6)))
   (* 0.5 (* re (+ (exp im) (exp (- im)))))
   (sin re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -9e-5) || !(im <= 6.2e-6)) {
		tmp = 0.5 * (re * (exp(im) + exp(-im)));
	} else {
		tmp = sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-9d-5)) .or. (.not. (im <= 6.2d-6))) then
        tmp = 0.5d0 * (re * (exp(im) + exp(-im)))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -9e-5) || !(im <= 6.2e-6)) {
		tmp = 0.5 * (re * (Math.exp(im) + Math.exp(-im)));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -9e-5) or not (im <= 6.2e-6):
		tmp = 0.5 * (re * (math.exp(im) + math.exp(-im)))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -9e-5) || !(im <= 6.2e-6))
		tmp = Float64(0.5 * Float64(re * Float64(exp(im) + exp(Float64(-im)))));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -9e-5) || ~((im <= 6.2e-6)))
		tmp = 0.5 * (re * (exp(im) + exp(-im)));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -9e-5], N[Not[LessEqual[im, 6.2e-6]], $MachinePrecision]], N[(0.5 * N[(re * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -9.00000000000000057e-5 or 6.1999999999999999e-6 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 68.9%

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

   if -9.00000000000000057e-5 < im < 6.1999999999999999e-6

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 99.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{-5} \lor \neg \left(im \leq 6.2 \cdot 10^{-6}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 3: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \mathbf{if}\;im \leq -5.8 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1900000000000:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (* re (* im im)) (* re 2.0)))))
   (if (<= im -5.8e+125)
     t_0
     (if (<= im -1900000000000.0)
       (* 0.5 (log1p (expm1 re)))
       (if (<= im 2e+20)
         (sin re)
         (if (<= im 4.5e+94) (* 0.5 (pow re -2.0)) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * im)) + (re * 2.0));
	double tmp;
	if (im <= -5.8e+125) {
		tmp = t_0;
	} else if (im <= -1900000000000.0) {
		tmp = 0.5 * log1p(expm1(re));
	} else if (im <= 2e+20) {
		tmp = sin(re);
	} else if (im <= 4.5e+94) {
		tmp = 0.5 * pow(re, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * im)) + (re * 2.0));
	double tmp;
	if (im <= -5.8e+125) {
		tmp = t_0;
	} else if (im <= -1900000000000.0) {
		tmp = 0.5 * Math.log1p(Math.expm1(re));
	} else if (im <= 2e+20) {
		tmp = Math.sin(re);
	} else if (im <= 4.5e+94) {
		tmp = 0.5 * Math.pow(re, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((re * (im * im)) + (re * 2.0))
	tmp = 0
	if im <= -5.8e+125:
		tmp = t_0
	elif im <= -1900000000000.0:
		tmp = 0.5 * math.log1p(math.expm1(re))
	elif im <= 2e+20:
		tmp = math.sin(re)
	elif im <= 4.5e+94:
		tmp = 0.5 * math.pow(re, -2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(re * Float64(im * im)) + Float64(re * 2.0)))
	tmp = 0.0
	if (im <= -5.8e+125)
		tmp = t_0;
	elseif (im <= -1900000000000.0)
		tmp = Float64(0.5 * log1p(expm1(re)));
	elseif (im <= 2e+20)
		tmp = sin(re);
	elseif (im <= 4.5e+94)
		tmp = Float64(0.5 * (re ^ -2.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.8e+125], t$95$0, If[LessEqual[im, -1900000000000.0], N[(0.5 * N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+20], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.5e+94], N[(0.5 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.79999999999999986e125 or 4.49999999999999972e94 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 67.5%

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

   3. Taylor expanded in im around 0 52.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + 2 \cdot re\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 23.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot {im}^{2}\right)\right)} + 2 \cdot re\right) \]

      2. expm1-udef 23.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot {im}^{2}\right)} - 1\right)} + 2 \cdot re\right) \]

      3. log1p-udef 23.0%

         \[\leadsto 0.5 \cdot \left(\left(e^{\color{blue}{\log \left(1 + re \cdot {im}^{2}\right)}} - 1\right) + 2 \cdot re\right) \]

      4. add-exp-log 52.1%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(1 + re \cdot {im}^{2}\right)} - 1\right) + 2 \cdot re\right) \]

      5. unpow2 52.1%

         \[\leadsto 0.5 \cdot \left(\left(\left(1 + re \cdot \color{blue}{\left(im \cdot im\right)}\right) - 1\right) + 2 \cdot re\right) \]

   5. Applied egg-rr 52.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(1 + re \cdot \left(im \cdot im\right)\right) - 1\right)} + 2 \cdot re\right) \]
   6. Step-by-step derivation

      1. +-commutative 52.1%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + 1\right)} - 1\right) + 2 \cdot re\right) \]

      2. associate--l+ 52.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + \left(1 - 1\right)\right)} + 2 \cdot re\right) \]

      3. metadata-eval 52.1%

         \[\leadsto 0.5 \cdot \left(\left(re \cdot \left(im \cdot im\right) + \color{blue}{0}\right) + 2 \cdot re\right) \]

      4. +-rgt-identity 52.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

   7. Simplified 52.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

   if -5.79999999999999986e125 < im < -1.9e12

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 70.4%

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

   4. Applied egg-rr 41.4%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]

   if -1.9e12 < im < 2e20

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 94.3%

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

   if 2e20 < im < 4.49999999999999972e94

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 68.8%

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

   4. Applied egg-rr 39.0%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{-2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \mathbf{elif}\;im \leq -1900000000000:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \end{array} \]

Alternative 4: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \mathbf{if}\;im \leq -0.000195:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (* re (* im im)) (* re 2.0)))))
   (if (<= im -0.000195)
     t_0
     (if (<= im 3.9e+18)
       (sin re)
       (if (<= im 2.2e+95) (* 0.5 (pow re -2.0)) t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * im)) + (re * 2.0));
	double tmp;
	if (im <= -0.000195) {
		tmp = t_0;
	} else if (im <= 3.9e+18) {
		tmp = sin(re);
	} else if (im <= 2.2e+95) {
		tmp = 0.5 * pow(re, -2.0);
	} else {
		tmp = 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 = 0.5d0 * ((re * (im * im)) + (re * 2.0d0))
    if (im <= (-0.000195d0)) then
        tmp = t_0
    else if (im <= 3.9d+18) then
        tmp = sin(re)
    else if (im <= 2.2d+95) then
        tmp = 0.5d0 * (re ** (-2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * im)) + (re * 2.0));
	double tmp;
	if (im <= -0.000195) {
		tmp = t_0;
	} else if (im <= 3.9e+18) {
		tmp = Math.sin(re);
	} else if (im <= 2.2e+95) {
		tmp = 0.5 * Math.pow(re, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((re * (im * im)) + (re * 2.0))
	tmp = 0
	if im <= -0.000195:
		tmp = t_0
	elif im <= 3.9e+18:
		tmp = math.sin(re)
	elif im <= 2.2e+95:
		tmp = 0.5 * math.pow(re, -2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(re * Float64(im * im)) + Float64(re * 2.0)))
	tmp = 0.0
	if (im <= -0.000195)
		tmp = t_0;
	elseif (im <= 3.9e+18)
		tmp = sin(re);
	elseif (im <= 2.2e+95)
		tmp = Float64(0.5 * (re ^ -2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((re * (im * im)) + (re * 2.0));
	tmp = 0.0;
	if (im <= -0.000195)
		tmp = t_0;
	elseif (im <= 3.9e+18)
		tmp = sin(re);
	elseif (im <= 2.2e+95)
		tmp = 0.5 * (re ^ -2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.000195], t$95$0, If[LessEqual[im, 3.9e+18], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.2e+95], N[(0.5 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.94999999999999996e-4 or 2.1999999999999999e95 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 68.2%

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

   3. Taylor expanded in im around 0 43.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + 2 \cdot re\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 19.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot {im}^{2}\right)\right)} + 2 \cdot re\right) \]

      2. expm1-udef 18.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot {im}^{2}\right)} - 1\right)} + 2 \cdot re\right) \]

      3. log1p-udef 18.6%

         \[\leadsto 0.5 \cdot \left(\left(e^{\color{blue}{\log \left(1 + re \cdot {im}^{2}\right)}} - 1\right) + 2 \cdot re\right) \]

      4. add-exp-log 42.6%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(1 + re \cdot {im}^{2}\right)} - 1\right) + 2 \cdot re\right) \]

      5. unpow2 42.6%

         \[\leadsto 0.5 \cdot \left(\left(\left(1 + re \cdot \color{blue}{\left(im \cdot im\right)}\right) - 1\right) + 2 \cdot re\right) \]

   5. Applied egg-rr 42.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(1 + re \cdot \left(im \cdot im\right)\right) - 1\right)} + 2 \cdot re\right) \]
   6. Step-by-step derivation

      1. +-commutative 42.6%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + 1\right)} - 1\right) + 2 \cdot re\right) \]

      2. associate--l+ 43.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + \left(1 - 1\right)\right)} + 2 \cdot re\right) \]

      3. metadata-eval 43.0%

         \[\leadsto 0.5 \cdot \left(\left(re \cdot \left(im \cdot im\right) + \color{blue}{0}\right) + 2 \cdot re\right) \]

      4. +-rgt-identity 43.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

   7. Simplified 43.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

   if -1.94999999999999996e-4 < im < 3.9e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 95.9%

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

   if 3.9e18 < im < 2.1999999999999999e95

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 68.8%

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

   4. Applied egg-rr 39.0%

      \[\leadsto 0.5 \cdot \color{blue}{{re}^{-2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.000195:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \end{array} \]

Alternative 5: 71.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00014 \lor \neg \left(im \leq 8.2 \cdot 10^{+79}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.00014) (not (<= im 8.2e+79)))
   (* 0.5 (+ (* re (* im im)) (* re 2.0)))
   (sin re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.00014) || !(im <= 8.2e+79)) {
		tmp = 0.5 * ((re * (im * im)) + (re * 2.0));
	} else {
		tmp = sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.00014d0)) .or. (.not. (im <= 8.2d+79))) then
        tmp = 0.5d0 * ((re * (im * im)) + (re * 2.0d0))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.00014) || !(im <= 8.2e+79)) {
		tmp = 0.5 * ((re * (im * im)) + (re * 2.0));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.00014) or not (im <= 8.2e+79):
		tmp = 0.5 * ((re * (im * im)) + (re * 2.0))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.00014) || !(im <= 8.2e+79))
		tmp = Float64(0.5 * Float64(Float64(re * Float64(im * im)) + Float64(re * 2.0)));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.00014) || ~((im <= 8.2e+79)))
		tmp = 0.5 * ((re * (im * im)) + (re * 2.0));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.00014], N[Not[LessEqual[im, 8.2e+79]], $MachinePrecision]], N[(0.5 * N[(N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.3999999999999999e-4 or 8.2e79 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 69.0%

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

   3. Taylor expanded in im around 0 41.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + 2 \cdot re\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 18.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot {im}^{2}\right)\right)} + 2 \cdot re\right) \]

      2. expm1-udef 18.2%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot {im}^{2}\right)} - 1\right)} + 2 \cdot re\right) \]

      3. log1p-udef 18.2%

         \[\leadsto 0.5 \cdot \left(\left(e^{\color{blue}{\log \left(1 + re \cdot {im}^{2}\right)}} - 1\right) + 2 \cdot re\right) \]

      4. add-exp-log 41.5%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(1 + re \cdot {im}^{2}\right)} - 1\right) + 2 \cdot re\right) \]

      5. unpow2 41.5%

         \[\leadsto 0.5 \cdot \left(\left(\left(1 + re \cdot \color{blue}{\left(im \cdot im\right)}\right) - 1\right) + 2 \cdot re\right) \]

   5. Applied egg-rr 41.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(1 + re \cdot \left(im \cdot im\right)\right) - 1\right)} + 2 \cdot re\right) \]
   6. Step-by-step derivation

      1. +-commutative 41.5%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + 1\right)} - 1\right) + 2 \cdot re\right) \]

      2. associate--l+ 41.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + \left(1 - 1\right)\right)} + 2 \cdot re\right) \]

      3. metadata-eval 41.9%

         \[\leadsto 0.5 \cdot \left(\left(re \cdot \left(im \cdot im\right) + \color{blue}{0}\right) + 2 \cdot re\right) \]

      4. +-rgt-identity 41.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

   7. Simplified 41.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

   if -1.3999999999999999e-4 < im < 8.2e79

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 87.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00014 \lor \neg \left(im \leq 8.2 \cdot 10^{+79}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 6: 48.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ (* re (* im im)) (* re 2.0))))

C

double code(double re, double im) {
	return 0.5 * ((re * (im * im)) + (re * 2.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((re * (im * im)) + (re * 2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((re * (im * im)) + (re * 2.0));
}

Python

def code(re, im):
	return 0.5 * ((re * (im * im)) + (re * 2.0))

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(re * Float64(im * im)) + Float64(re * 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((re * (im * im)) + (re * 2.0));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0 61.3%

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

3. Taylor expanded in im around 0 44.8%

   \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + 2 \cdot re\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 34.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot {im}^{2}\right)\right)} + 2 \cdot re\right) \]

   2. expm1-udef 34.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot {im}^{2}\right)} - 1\right)} + 2 \cdot re\right) \]

   3. log1p-udef 34.1%

      \[\leadsto 0.5 \cdot \left(\left(e^{\color{blue}{\log \left(1 + re \cdot {im}^{2}\right)}} - 1\right) + 2 \cdot re\right) \]

   4. add-exp-log 44.5%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(1 + re \cdot {im}^{2}\right)} - 1\right) + 2 \cdot re\right) \]

   5. unpow2 44.5%

      \[\leadsto 0.5 \cdot \left(\left(\left(1 + re \cdot \color{blue}{\left(im \cdot im\right)}\right) - 1\right) + 2 \cdot re\right) \]

5. Applied egg-rr 44.5%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(1 + re \cdot \left(im \cdot im\right)\right) - 1\right)} + 2 \cdot re\right) \]
6. Step-by-step derivation

   1. +-commutative 44.5%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + 1\right)} - 1\right) + 2 \cdot re\right) \]

   2. associate--l+ 44.8%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot \left(im \cdot im\right) + \left(1 - 1\right)\right)} + 2 \cdot re\right) \]

   3. metadata-eval 44.8%

      \[\leadsto 0.5 \cdot \left(\left(re \cdot \left(im \cdot im\right) + \color{blue}{0}\right) + 2 \cdot re\right) \]

   4. +-rgt-identity 44.8%

      \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

7. Simplified 44.8%

   \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot im\right)} + 2 \cdot re\right) \]

8. Final simplification 44.8%

   \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot im\right) + re \cdot 2\right) \]

Alternative 7: 26.5% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0 61.3%

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

4. Applied egg-rr 27.5%

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

5. Final simplification 27.5%

   \[\leadsto 0.5 \cdot \left(re + re\right) \]

Alternative 8: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -2.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -2.0

Julia

function code(re, im)
	return -2.0
end

MATLAB

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

Wolfram

code[re_, im_] := -2.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. add-log-exp 76.8%

      \[\leadsto \color{blue}{\log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   2. *-un-lft-identity 76.8%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   3. log-prod 76.8%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   4. metadata-eval 76.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   6. +-commutative 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

   7. sub0-neg 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{\color{blue}{-im}}\right) \]

   8. cosh-undef 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \left(0.5 \cdot \sin re\right) \cdot \left(2 \cdot \cosh im\right)} \]

5. Applied egg-rr 4.8%

   \[\leadsto 0 + \color{blue}{-2} \]

6. Final simplification 4.8%

   \[\leadsto -2 \]

Alternative 9: 4.7% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. add-log-exp 76.8%

      \[\leadsto \color{blue}{\log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   2. *-un-lft-identity 76.8%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   3. log-prod 76.8%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right)} \]

   4. metadata-eval 76.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   6. +-commutative 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

   7. sub0-neg 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{\color{blue}{-im}}\right) \]

   8. cosh-undef 100.0%

      \[\leadsto 0 + \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \left(0.5 \cdot \sin re\right) \cdot \left(2 \cdot \cosh im\right)} \]

5. Applied egg-rr 5.3%

   \[\leadsto 0 + \color{blue}{-1} \]

6. Final simplification 5.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))