math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 7

Speedup: 1.0×

• 49.2% 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, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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(-im) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.06 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.06) (not (<= im 1.35e+154)))
   (+ (sin re) (* (* 0.5 (sin re)) (* im im)))
   (* (* 0.5 re) (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.06) || !(im <= 1.35e+154)) {
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	} else {
		tmp = (0.5 * re) * (exp(-im) + exp(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 <= 0.06d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = sin(re) + ((0.5d0 * sin(re)) * (im * im))
    else
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.06) || !(im <= 1.35e+154)) {
		tmp = Math.sin(re) + ((0.5 * Math.sin(re)) * (im * im));
	} else {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.06) or not (im <= 1.35e+154):
		tmp = math.sin(re) + ((0.5 * math.sin(re)) * (im * im))
	else:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.06) || !(im <= 1.35e+154))
		tmp = Float64(sin(re) + Float64(Float64(0.5 * sin(re)) * Float64(im * im)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.06) || ~((im <= 1.35e+154)))
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	else
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.06], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.059999999999999998 or 1.35000000000000003e154 < im

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.5%

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

   6. Simplified 85.5%

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

   if 0.059999999999999998 < im < 1.35000000000000003e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.06 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 3: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00019:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00019) (sin re) (* (* 0.5 re) (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00019) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * (exp(-im) + exp(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 <= 0.00019d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00019) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00019:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00019)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00019)
		tmp = sin(re);
	else
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00019], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.9000000000000001e-4

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.0%

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

   if 1.9000000000000001e-4 < im

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 83.7%

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

   6. Simplified 83.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00019:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 4: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4600000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+71}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4600000000.0)
   (sin re)
   (if (<= im 1.45e+71) (pow re -512.0) (+ re (* 0.5 (* re (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4600000000.0) {
		tmp = sin(re);
	} else if (im <= 1.45e+71) {
		tmp = pow(re, -512.0);
	} else {
		tmp = re + (0.5 * (re * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4600000000.0d0) then
        tmp = sin(re)
    else if (im <= 1.45d+71) then
        tmp = re ** (-512.0d0)
    else
        tmp = re + (0.5d0 * (re * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4600000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.45e+71) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = re + (0.5 * (re * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4600000000.0:
		tmp = math.sin(re)
	elif im <= 1.45e+71:
		tmp = math.pow(re, -512.0)
	else:
		tmp = re + (0.5 * (re * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4600000000.0)
		tmp = sin(re);
	elseif (im <= 1.45e+71)
		tmp = re ^ -512.0;
	else
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4600000000.0)
		tmp = sin(re);
	elseif (im <= 1.45e+71)
		tmp = re ^ -512.0;
	else
		tmp = re + (0.5 * (re * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4600000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.45e+71], N[Power[re, -512.0], $MachinePrecision], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.6e9

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.7%

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

   if 4.6e9 < im < 1.45000000000000004e71

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.0%

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

   6. Simplified 75.0%

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

   8. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 1.45000000000000004e71 < im

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in im around 0 70.2%

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

      1. add-log-exp 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. log-prod 83.7%

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

      4. metadata-eval 83.7%

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

      5. add-log-exp 70.2%

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

      6. pow2 70.2%

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

      7. *-commutative 70.2%

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

      8. associate-*l* 54.7%

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

   9. Applied egg-rr 54.7%

      \[\leadsto re + 0.5 \cdot \color{blue}{\left(0 + im \cdot \left(im \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. +-lft-identity 54.7%

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

       2. associate-*r* 70.2%

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

       3. *-commutative 70.2%

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

   11. Simplified 70.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4600000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+71}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 62.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.031:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.031) (sin re) (+ re (* 0.5 (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.031) {
		tmp = sin(re);
	} else {
		tmp = re + (0.5 * (re * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.031d0) then
        tmp = sin(re)
    else
        tmp = re + (0.5d0 * (re * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.031) {
		tmp = Math.sin(re);
	} else {
		tmp = re + (0.5 * (re * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.031:
		tmp = math.sin(re)
	else:
		tmp = re + (0.5 * (re * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.031)
		tmp = sin(re);
	else
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.031)
		tmp = sin(re);
	else
		tmp = re + (0.5 * (re * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.031], N[Sin[re], $MachinePrecision], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.031

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.7%

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

   if 0.031 < im

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 84.6%

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

   6. Simplified 84.6%

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

   7. Taylor expanded in im around 0 53.5%

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

      1. add-log-exp 72.5%

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

      2. *-un-lft-identity 72.5%

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

      3. log-prod 72.5%

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

      4. metadata-eval 72.5%

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

      5. add-log-exp 53.5%

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

      6. pow2 53.5%

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

      7. *-commutative 53.5%

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

      8. associate-*l* 41.8%

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

   9. Applied egg-rr 41.8%

      \[\leadsto re + 0.5 \cdot \color{blue}{\left(0 + im \cdot \left(im \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. +-lft-identity 41.8%

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

       2. associate-*r* 53.5%

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

       3. *-commutative 53.5%

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

   11. Simplified 53.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.031:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 49.2% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in im around 0 52.5%

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

   1. add-log-exp 60.4%

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

   2. *-un-lft-identity 60.4%

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

   3. log-prod 60.4%

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

   4. metadata-eval 60.4%

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

   5. add-log-exp 52.5%

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

   6. pow2 52.5%

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

   7. *-commutative 52.5%

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

   8. associate-*l* 46.2%

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

9. Applied egg-rr 46.2%

   \[\leadsto re + 0.5 \cdot \color{blue}{\left(0 + im \cdot \left(im \cdot re\right)\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 46.2%

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

    2. associate-*r* 52.5%

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

    3. *-commutative 52.5%

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

11. Simplified 52.5%

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

12. Final simplification 52.5%

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

Alternative 7: 27.4% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in im around 0 28.5%

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

8. Final simplification 28.5%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023200 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))