math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 9

Speedup: 1.0×

• 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, 1.0× 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: 88.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154} \lor \neg \left(im \leq -2.6 \lor \neg \left(im \leq 1.95\right) \land im \leq 1.2 \cdot 10^{+247}\right):\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.35e+154)
         (not (or (<= im -2.6) (and (not (<= im 1.95)) (<= im 1.2e+247)))))
   (+ (sin re) (* 0.5 (* im (* (sin re) im))))
   (* (* 0.5 re) (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.35e+154) || !((im <= -2.6) || (!(im <= 1.95) && (im <= 1.2e+247)))) {
		tmp = sin(re) + (0.5 * (im * (sin(re) * 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 <= (-1.35d+154)) .or. (.not. (im <= (-2.6d0)) .or. (.not. (im <= 1.95d0)) .and. (im <= 1.2d+247))) then
        tmp = sin(re) + (0.5d0 * (im * (sin(re) * 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 <= -1.35e+154) || !((im <= -2.6) || (!(im <= 1.95) && (im <= 1.2e+247)))) {
		tmp = Math.sin(re) + (0.5 * (im * (Math.sin(re) * im)));
	} else {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.35e+154) or not ((im <= -2.6) or (not (im <= 1.95) and (im <= 1.2e+247))):
		tmp = math.sin(re) + (0.5 * (im * (math.sin(re) * im)))
	else:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.35e+154) || !((im <= -2.6) || (!(im <= 1.95) && (im <= 1.2e+247))))
		tmp = Float64(sin(re) + Float64(0.5 * Float64(im * Float64(sin(re) * 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 <= -1.35e+154) || ~(((im <= -2.6) || (~((im <= 1.95)) && (im <= 1.2e+247)))))
		tmp = sin(re) + (0.5 * (im * (sin(re) * im)));
	else
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.35e+154], N[Not[Or[LessEqual[im, -2.6], And[N[Not[LessEqual[im, 1.95]], $MachinePrecision], LessEqual[im, 1.2e+247]]]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $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 < -1.35000000000000003e154 or -2.60000000000000009 < im < 1.94999999999999996 or 1.2e247 < 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 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in re around inf 99.1%

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

   9. Simplified 95.8%

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

   if -1.35000000000000003e154 < im < -2.60000000000000009 or 1.94999999999999996 < im < 1.2e247

   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 74.7%

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

   6. Simplified 74.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154} \lor \neg \left(im \leq -2.6 \lor \neg \left(im \leq 1.95\right) \land im \leq 1.2 \cdot 10^{+247}\right):\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 3: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 re) (+ (exp (- im)) (exp im))))
        (t_1 (+ (sin re) (* (* 0.5 (sin re)) (* im im)))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -2.6)
       t_0
       (if (<= im 1.95)
         (+ (sin re) (* 0.5 (* im (* (sin re) im))))
         (if (<= im 1.4e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * re) * (exp(-im) + exp(im));
	double t_1 = sin(re) + ((0.5 * sin(re)) * (im * im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -2.6) {
		tmp = t_0;
	} else if (im <= 1.95) {
		tmp = sin(re) + (0.5 * (im * (sin(re) * im)));
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * re) * (exp(-im) + exp(im))
    t_1 = sin(re) + ((0.5d0 * sin(re)) * (im * im))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-2.6d0)) then
        tmp = t_0
    else if (im <= 1.95d0) then
        tmp = sin(re) + (0.5d0 * (im * (sin(re) * im)))
    else if (im <= 1.4d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	double t_1 = Math.sin(re) + ((0.5 * Math.sin(re)) * (im * im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -2.6) {
		tmp = t_0;
	} else if (im <= 1.95) {
		tmp = Math.sin(re) + (0.5 * (im * (Math.sin(re) * im)));
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * re) * (math.exp(-im) + math.exp(im))
	t_1 = math.sin(re) + ((0.5 * math.sin(re)) * (im * im))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -2.6:
		tmp = t_0
	elif im <= 1.95:
		tmp = math.sin(re) + (0.5 * (im * (math.sin(re) * im)))
	elif im <= 1.4e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = Float64(sin(re) + Float64(Float64(0.5 * sin(re)) * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -2.6)
		tmp = t_0;
	elseif (im <= 1.95)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(im * Float64(sin(re) * im))));
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * re) * (exp(-im) + exp(im));
	t_1 = sin(re) + ((0.5 * sin(re)) * (im * im));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -2.6)
		tmp = t_0;
	elseif (im <= 1.95)
		tmp = sin(re) + (0.5 * (im * (sin(re) * im)));
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -2.6], t$95$0, If[LessEqual[im, 1.95], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or 1.4e154 < 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 0 100.0%

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

   6. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -2.60000000000000009 or 1.94999999999999996 < im < 1.4e154

   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 73.0%

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

   6. Simplified 73.0%

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

   if -2.60000000000000009 < im < 1.94999999999999996

   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 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in re around inf 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 4: 87.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.000195 \lor \neg \left(im \leq 6.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.000195) (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 <= -0.000195) || !(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 <= (-0.000195d0)) .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 <= -0.000195) || !(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 <= -0.000195) 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 <= -0.000195) || !(im <= 6.2e-6))
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.000195) || ~((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, -0.000195], N[Not[LessEqual[im, 6.2e-6]], $MachinePrecision]], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.94999999999999996e-4 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. 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 68.9%

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

   6. Simplified 68.9%

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

   if -1.94999999999999996e-4 < 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. 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 99.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.000195 \lor \neg \left(im \leq 6.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + 0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \mathbf{if}\;im \leq -5.9 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+94}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* 0.5 (* re (pow im 2.0))))))
   (if (<= im -5.9e+125)
     t_0
     (if (<= im -2300000000000.0)
       (log1p (expm1 re))
       (if (<= im 4e+18) (sin re) (if (<= im 9.5e+94) (pow re -512.0) t_0))))))

C

double code(double re, double im) {
	double t_0 = re + (0.5 * (re * pow(im, 2.0)));
	double tmp;
	if (im <= -5.9e+125) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = log1p(expm1(re));
	} else if (im <= 4e+18) {
		tmp = sin(re);
	} else if (im <= 9.5e+94) {
		tmp = pow(re, -512.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = re + (0.5 * (re * Math.pow(im, 2.0)));
	double tmp;
	if (im <= -5.9e+125) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = Math.log1p(Math.expm1(re));
	} else if (im <= 4e+18) {
		tmp = Math.sin(re);
	} else if (im <= 9.5e+94) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (0.5 * (re * math.pow(im, 2.0)))
	tmp = 0
	if im <= -5.9e+125:
		tmp = t_0
	elif im <= -2300000000000.0:
		tmp = math.log1p(math.expm1(re))
	elif im <= 4e+18:
		tmp = math.sin(re)
	elif im <= 9.5e+94:
		tmp = math.pow(re, -512.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(0.5 * Float64(re * (im ^ 2.0))))
	tmp = 0.0
	if (im <= -5.9e+125)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = log1p(expm1(re));
	elseif (im <= 4e+18)
		tmp = sin(re);
	elseif (im <= 9.5e+94)
		tmp = re ^ -512.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.9e+125], t$95$0, If[LessEqual[im, -2300000000000.0], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 4e+18], N[Sin[re], $MachinePrecision], If[LessEqual[im, 9.5e+94], N[Power[re, -512.0], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.9000000000000001e125 or 9.4999999999999998e94 < 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 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 52.1%

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

   if -5.9000000000000001e125 < im < -2.3e12

   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 70.4%

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

   6. Simplified 70.4%

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

   8. Applied egg-rr 41.4%

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

   if -2.3e12 < im < 4e18

   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 94.3%

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

   if 4e18 < im < 9.4999999999999998e94

   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 68.8%

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

   6. Simplified 68.8%

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

   8. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{{re}^{-512}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.9 \cdot 10^{+125}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+94}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]

Alternative 6: 73.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + 0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \mathbf{if}\;im \leq -0.00019:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* 0.5 (* re (pow im 2.0))))))
   (if (<= im -0.00019)
     t_0
     (if (<= im 3.9e+18) (sin re) (if (<= im 1.8e+95) (pow re -512.0) t_0)))))

C

double code(double re, double im) {
	double t_0 = re + (0.5 * (re * pow(im, 2.0)));
	double tmp;
	if (im <= -0.00019) {
		tmp = t_0;
	} else if (im <= 3.9e+18) {
		tmp = sin(re);
	} else if (im <= 1.8e+95) {
		tmp = pow(re, -512.0);
	} 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 = re + (0.5d0 * (re * (im ** 2.0d0)))
    if (im <= (-0.00019d0)) then
        tmp = t_0
    else if (im <= 3.9d+18) then
        tmp = sin(re)
    else if (im <= 1.8d+95) then
        tmp = re ** (-512.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + (0.5 * (re * Math.pow(im, 2.0)));
	double tmp;
	if (im <= -0.00019) {
		tmp = t_0;
	} else if (im <= 3.9e+18) {
		tmp = Math.sin(re);
	} else if (im <= 1.8e+95) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (0.5 * (re * math.pow(im, 2.0)))
	tmp = 0
	if im <= -0.00019:
		tmp = t_0
	elif im <= 3.9e+18:
		tmp = math.sin(re)
	elif im <= 1.8e+95:
		tmp = math.pow(re, -512.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(0.5 * Float64(re * (im ^ 2.0))))
	tmp = 0.0
	if (im <= -0.00019)
		tmp = t_0;
	elseif (im <= 3.9e+18)
		tmp = sin(re);
	elseif (im <= 1.8e+95)
		tmp = re ^ -512.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + (0.5 * (re * (im ^ 2.0)));
	tmp = 0.0;
	if (im <= -0.00019)
		tmp = t_0;
	elseif (im <= 3.9e+18)
		tmp = sin(re);
	elseif (im <= 1.8e+95)
		tmp = re ^ -512.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.00019], t$95$0, If[LessEqual[im, 3.9e+18], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.8e+95], N[Power[re, -512.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.9000000000000001e-4 or 1.79999999999999989e95 < 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 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in im around 0 43.0%

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

   if -1.9000000000000001e-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. 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 95.9%

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

   if 3.9e18 < im < 1.79999999999999989e95

   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 68.8%

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

   6. Simplified 68.8%

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

   8. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{{re}^{-512}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00019:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]

Alternative 7: 62.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-512}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.6e+18)
   (pow re -512.0)
   (if (<= im 3.9e+18) (sin re) (pow re -512.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.6e+18) {
		tmp = pow(re, -512.0);
	} else if (im <= 3.9e+18) {
		tmp = sin(re);
	} else {
		tmp = pow(re, -512.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.6d+18)) then
        tmp = re ** (-512.0d0)
    else if (im <= 3.9d+18) then
        tmp = sin(re)
    else
        tmp = re ** (-512.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.6e+18) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 3.9e+18) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.pow(re, -512.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.6e+18:
		tmp = math.pow(re, -512.0)
	elif im <= 3.9e+18:
		tmp = math.sin(re)
	else:
		tmp = math.pow(re, -512.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.6e+18)
		tmp = re ^ -512.0;
	elseif (im <= 3.9e+18)
		tmp = sin(re);
	else
		tmp = re ^ -512.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.6e+18)
		tmp = re ^ -512.0;
	elseif (im <= 3.9e+18)
		tmp = sin(re);
	else
		tmp = re ^ -512.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.6e+18], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 3.9e+18], N[Sin[re], $MachinePrecision], N[Power[re, -512.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < -1.6e18 or 3.9e18 < 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 68.6%

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

   6. Simplified 68.6%

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

   8. Applied egg-rr 21.5%

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

   if -1.6e18 < im < 3.9e18

   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 93.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-512}\\ \end{array} \]

Alternative 8: 51.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sin(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sin[re], $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 im around 0 50.3%

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

5. Final simplification 50.3%

   \[\leadsto \sin re \]

Alternative 9: 26.5% 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 61.3%

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

6. Simplified 61.3%

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

7. Taylor expanded in im around 0 27.5%

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

8. Final simplification 27.5%

   \[\leadsto re \]

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))))