math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 15

Speedup: 1.0×

• 49.8% 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

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((0.5 * Math.exp(im)) + (0.5 * (1.0 / Math.exp(im))));
}

Python

def code(re, im):
	return math.sin(re) * ((0.5 * math.exp(im)) + (0.5 * (1.0 / math.exp(im))))

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $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. distribute-rgt-in 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

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

   9. *-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-commutative 100.0%

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

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

4. Taylor expanded in re around inf 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \frac{1}{e^{im}}\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $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. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

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

   8. associate-*r* 100.0%

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

   9. distribute-rgt-out-- 100.0%

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

   10. sub-neg 100.0%

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

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

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%

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

Alternative 3: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return sin(re) * fma(0.5, exp(im), 0.5);
}

Julia

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

Wolfram

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

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

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

   9. *-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-commutative 100.0%

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

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

4. Taylor expanded in im around 0 75.5%

   \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.5}\right) \]

5. Final simplification 75.5%

   \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, 0.5\right) \]

Alternative 4: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.3 \cdot 10^{+105}:\\ \;\;\;\;-0.75 \cdot {re}^{3} + re \cdot 4.5\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ (* 0.5 (exp im)) 4.0))))
   (if (<= im 125.0)
     (sin re)
     (if (<= im 1.55e+67)
       t_0
       (if (<= im 6.3e+105)
         (+ (* -0.75 (pow re 3.0)) (* re 4.5))
         (if (<= im 1.32e+154) t_0 (* (* (sin re) 0.5) (pow im 2.0))))))))

C

double code(double re, double im) {
	double t_0 = re * ((0.5 * exp(im)) + 4.0);
	double tmp;
	if (im <= 125.0) {
		tmp = sin(re);
	} else if (im <= 1.55e+67) {
		tmp = t_0;
	} else if (im <= 6.3e+105) {
		tmp = (-0.75 * pow(re, 3.0)) + (re * 4.5);
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = (sin(re) * 0.5) * pow(im, 2.0);
	}
	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 * exp(im)) + 4.0d0)
    if (im <= 125.0d0) then
        tmp = sin(re)
    else if (im <= 1.55d+67) then
        tmp = t_0
    else if (im <= 6.3d+105) then
        tmp = ((-0.75d0) * (re ** 3.0d0)) + (re * 4.5d0)
    else if (im <= 1.32d+154) then
        tmp = t_0
    else
        tmp = (sin(re) * 0.5d0) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((0.5 * Math.exp(im)) + 4.0);
	double tmp;
	if (im <= 125.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.55e+67) {
		tmp = t_0;
	} else if (im <= 6.3e+105) {
		tmp = (-0.75 * Math.pow(re, 3.0)) + (re * 4.5);
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = (Math.sin(re) * 0.5) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((0.5 * math.exp(im)) + 4.0)
	tmp = 0
	if im <= 125.0:
		tmp = math.sin(re)
	elif im <= 1.55e+67:
		tmp = t_0
	elif im <= 6.3e+105:
		tmp = (-0.75 * math.pow(re, 3.0)) + (re * 4.5)
	elif im <= 1.32e+154:
		tmp = t_0
	else:
		tmp = (math.sin(re) * 0.5) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64(0.5 * exp(im)) + 4.0))
	tmp = 0.0
	if (im <= 125.0)
		tmp = sin(re);
	elseif (im <= 1.55e+67)
		tmp = t_0;
	elseif (im <= 6.3e+105)
		tmp = Float64(Float64(-0.75 * (re ^ 3.0)) + Float64(re * 4.5));
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(sin(re) * 0.5) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((0.5 * exp(im)) + 4.0);
	tmp = 0.0;
	if (im <= 125.0)
		tmp = sin(re);
	elseif (im <= 1.55e+67)
		tmp = t_0;
	elseif (im <= 6.3e+105)
		tmp = (-0.75 * (re ^ 3.0)) + (re * 4.5);
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = (sin(re) * 0.5) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 125.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.55e+67], t$95$0, If[LessEqual[im, 6.3e+105], N[(N[(-0.75 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] + N[(re * 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.32e+154], t$95$0, N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 125

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.7%

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

   if 125 < im < 1.54999999999999998e67 or 6.29999999999999953e105 < im < 1.31999999999999998e154

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around 0 94.7%

      \[\leadsto \color{blue}{re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]

   if 1.54999999999999998e67 < im < 6.29999999999999953e105

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 2.9%

      \[\leadsto \color{blue}{4.5 \cdot \sin re} \]
   8. Step-by-step derivation

      1. *-commutative 2.9%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} \]

   9. Simplified 2.9%

      \[\leadsto \color{blue}{\sin re \cdot 4.5} \]

   10. Taylor expanded in re around 0 67.0%

       \[\leadsto \color{blue}{-0.75 \cdot {re}^{3} + 4.5 \cdot re} \]

   if 1.31999999999999998e154 < 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. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. sub-neg 100.0%

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

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   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%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   7. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \sin re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+67}:\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{elif}\;im \leq 6.3 \cdot 10^{+105}:\\ \;\;\;\;-0.75 \cdot {re}^{3} + re \cdot 4.5\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.6:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.6) (sin re) (* (sin re) (+ (* 0.5 (exp im)) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.6) {
		tmp = sin(re);
	} else {
		tmp = sin(re) * ((0.5 * exp(im)) + 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.6d0) then
        tmp = sin(re)
    else
        tmp = sin(re) * ((0.5d0 * exp(im)) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.6) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.sin(re) * ((0.5 * Math.exp(im)) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.6:
		tmp = math.sin(re)
	else:
		tmp = math.sin(re) * ((0.5 * math.exp(im)) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.6)
		tmp = sin(re);
	else
		tmp = Float64(sin(re) * Float64(Float64(0.5 * exp(im)) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.6)
		tmp = sin(re);
	else
		tmp = sin(re) * ((0.5 * exp(im)) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.6], N[Sin[re], $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.6000000000000001

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 69.0%

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

   if 1.6000000000000001 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \end{array} \]

Alternative 6: 87.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7)
   (* (* (sin re) 0.5) (fma im im 2.0))
   (* (sin re) (+ (* 0.5 (exp im)) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = (sin(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = sin(re) * ((0.5 * exp(im)) + 4.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.7)
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(sin(re) * Float64(Float64(0.5 * exp(im)) + 4.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.7], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.7000000000000002

   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. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. sub-neg 100.0%

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

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   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 83.6%

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

      1. +-commutative 83.6%

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

      2. unpow2 83.6%

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

      3. fma-def 83.6%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Simplified 83.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 2.7000000000000002 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \end{array} \]

Alternative 7: 70.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.3 \cdot 10^{+105}:\\ \;\;\;\;-0.75 \cdot {re}^{3} + re \cdot 4.5\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ (* 0.5 (exp im)) 4.0))))
   (if (<= im 125.0)
     (sin re)
     (if (<= im 1.55e+67)
       t_0
       (if (<= im 6.3e+105)
         (+ (* -0.75 (pow re 3.0)) (* re 4.5))
         (if (<= im 3e+154)
           t_0
           (* (sin re) (+ 4.5 (* im (+ 0.5 (* im 0.25)))))))))))

C

double code(double re, double im) {
	double t_0 = re * ((0.5 * exp(im)) + 4.0);
	double tmp;
	if (im <= 125.0) {
		tmp = sin(re);
	} else if (im <= 1.55e+67) {
		tmp = t_0;
	} else if (im <= 6.3e+105) {
		tmp = (-0.75 * pow(re, 3.0)) + (re * 4.5);
	} else if (im <= 3e+154) {
		tmp = t_0;
	} else {
		tmp = sin(re) * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	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 * exp(im)) + 4.0d0)
    if (im <= 125.0d0) then
        tmp = sin(re)
    else if (im <= 1.55d+67) then
        tmp = t_0
    else if (im <= 6.3d+105) then
        tmp = ((-0.75d0) * (re ** 3.0d0)) + (re * 4.5d0)
    else if (im <= 3d+154) then
        tmp = t_0
    else
        tmp = sin(re) * (4.5d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((0.5 * Math.exp(im)) + 4.0);
	double tmp;
	if (im <= 125.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.55e+67) {
		tmp = t_0;
	} else if (im <= 6.3e+105) {
		tmp = (-0.75 * Math.pow(re, 3.0)) + (re * 4.5);
	} else if (im <= 3e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(re) * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((0.5 * math.exp(im)) + 4.0)
	tmp = 0
	if im <= 125.0:
		tmp = math.sin(re)
	elif im <= 1.55e+67:
		tmp = t_0
	elif im <= 6.3e+105:
		tmp = (-0.75 * math.pow(re, 3.0)) + (re * 4.5)
	elif im <= 3e+154:
		tmp = t_0
	else:
		tmp = math.sin(re) * (4.5 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64(0.5 * exp(im)) + 4.0))
	tmp = 0.0
	if (im <= 125.0)
		tmp = sin(re);
	elseif (im <= 1.55e+67)
		tmp = t_0;
	elseif (im <= 6.3e+105)
		tmp = Float64(Float64(-0.75 * (re ^ 3.0)) + Float64(re * 4.5));
	elseif (im <= 3e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(re) * Float64(4.5 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((0.5 * exp(im)) + 4.0);
	tmp = 0.0;
	if (im <= 125.0)
		tmp = sin(re);
	elseif (im <= 1.55e+67)
		tmp = t_0;
	elseif (im <= 6.3e+105)
		tmp = (-0.75 * (re ^ 3.0)) + (re * 4.5);
	elseif (im <= 3e+154)
		tmp = t_0;
	else
		tmp = sin(re) * (4.5 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 125.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.55e+67], t$95$0, If[LessEqual[im, 6.3e+105], N[(N[(-0.75 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] + N[(re * 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e+154], t$95$0, N[(N[Sin[re], $MachinePrecision] * N[(4.5 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 125

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.7%

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

   if 125 < im < 1.54999999999999998e67 or 6.29999999999999953e105 < im < 3.00000000000000026e154

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around 0 94.7%

      \[\leadsto \color{blue}{re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]

   if 1.54999999999999998e67 < im < 6.29999999999999953e105

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 2.9%

      \[\leadsto \color{blue}{4.5 \cdot \sin re} \]
   8. Step-by-step derivation

      1. *-commutative 2.9%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} \]

   9. Simplified 2.9%

      \[\leadsto \color{blue}{\sin re \cdot 4.5} \]

   10. Taylor expanded in re around 0 67.0%

       \[\leadsto \color{blue}{-0.75 \cdot {re}^{3} + 4.5 \cdot re} \]

   if 3.00000000000000026e154 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.25 \cdot \left({im}^{2} \cdot \sin re\right) + \left(0.5 \cdot \left(im \cdot \sin re\right) + 4.5 \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) + 4.5 \cdot \sin re} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{4.5 \cdot \sin re + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \sin re \cdot 4.5 + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \sin re} + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \sin re \cdot 4.5 + \left(\left(0.25 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.5 \cdot im\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \sin re \cdot 4.5 + \color{blue}{\sin re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      8. unpow2 100.0%

         \[\leadsto \sin re \cdot \left(4.5 + \left(0.25 \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot im\right)\right) \]

      9. associate-*r* 100.0%

         \[\leadsto \sin re \cdot \left(4.5 + \left(\color{blue}{\left(0.25 \cdot im\right) \cdot im} + 0.5 \cdot im\right)\right) \]

      10. distribute-rgt-out 100.0%

          \[\leadsto \sin re \cdot \left(4.5 + \color{blue}{im \cdot \left(0.25 \cdot im + 0.5\right)}\right) \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + im \cdot \left(0.25 \cdot im + 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+67}:\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{elif}\;im \leq 6.3 \cdot 10^{+105}:\\ \;\;\;\;-0.75 \cdot {re}^{3} + re \cdot 4.5\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 8: 67.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+67} \lor \neg \left(im \leq 6.3 \cdot 10^{+105}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-0.75 \cdot {re}^{3} + re \cdot 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125.0)
   (sin re)
   (if (or (<= im 1.55e+67) (not (<= im 6.3e+105)))
     (* re (+ (* 0.5 (exp im)) 4.0))
     (+ (* -0.75 (pow re 3.0)) (* re 4.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = sin(re);
	} else if ((im <= 1.55e+67) || !(im <= 6.3e+105)) {
		tmp = re * ((0.5 * exp(im)) + 4.0);
	} else {
		tmp = (-0.75 * pow(re, 3.0)) + (re * 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 125.0d0) then
        tmp = sin(re)
    else if ((im <= 1.55d+67) .or. (.not. (im <= 6.3d+105))) then
        tmp = re * ((0.5d0 * exp(im)) + 4.0d0)
    else
        tmp = ((-0.75d0) * (re ** 3.0d0)) + (re * 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = Math.sin(re);
	} else if ((im <= 1.55e+67) || !(im <= 6.3e+105)) {
		tmp = re * ((0.5 * Math.exp(im)) + 4.0);
	} else {
		tmp = (-0.75 * Math.pow(re, 3.0)) + (re * 4.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 125.0:
		tmp = math.sin(re)
	elif (im <= 1.55e+67) or not (im <= 6.3e+105):
		tmp = re * ((0.5 * math.exp(im)) + 4.0)
	else:
		tmp = (-0.75 * math.pow(re, 3.0)) + (re * 4.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 125.0)
		tmp = sin(re);
	elseif ((im <= 1.55e+67) || !(im <= 6.3e+105))
		tmp = Float64(re * Float64(Float64(0.5 * exp(im)) + 4.0));
	else
		tmp = Float64(Float64(-0.75 * (re ^ 3.0)) + Float64(re * 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 125.0)
		tmp = sin(re);
	elseif ((im <= 1.55e+67) || ~((im <= 6.3e+105)))
		tmp = re * ((0.5 * exp(im)) + 4.0);
	else
		tmp = (-0.75 * (re ^ 3.0)) + (re * 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 125.0], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 1.55e+67], N[Not[LessEqual[im, 6.3e+105]], $MachinePrecision]], N[(re * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.75 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] + N[(re * 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 125

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.7%

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

   if 125 < im < 1.54999999999999998e67 or 6.29999999999999953e105 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]

   if 1.54999999999999998e67 < im < 6.29999999999999953e105

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 2.9%

      \[\leadsto \color{blue}{4.5 \cdot \sin re} \]
   8. Step-by-step derivation

      1. *-commutative 2.9%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} \]

   9. Simplified 2.9%

      \[\leadsto \color{blue}{\sin re \cdot 4.5} \]

   10. Taylor expanded in re around 0 67.0%

       \[\leadsto \color{blue}{-0.75 \cdot {re}^{3} + 4.5 \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+67} \lor \neg \left(im \leq 6.3 \cdot 10^{+105}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-0.75 \cdot {re}^{3} + re \cdot 4.5\\ \end{array} \]

Alternative 9: 64.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{{re}^{3}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 800.0)
   (sin re)
   (if (<= im 8.2e+138) (/ 1.0 (pow re 3.0)) (* re (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 800.0) {
		tmp = sin(re);
	} else if (im <= 8.2e+138) {
		tmp = 1.0 / pow(re, 3.0);
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 800.0d0) then
        tmp = sin(re)
    else if (im <= 8.2d+138) then
        tmp = 1.0d0 / (re ** 3.0d0)
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 800.0) {
		tmp = Math.sin(re);
	} else if (im <= 8.2e+138) {
		tmp = 1.0 / Math.pow(re, 3.0);
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 800.0:
		tmp = math.sin(re)
	elif im <= 8.2e+138:
		tmp = 1.0 / math.pow(re, 3.0)
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 800.0)
		tmp = sin(re);
	elseif (im <= 8.2e+138)
		tmp = Float64(1.0 / (re ^ 3.0));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 800.0)
		tmp = sin(re);
	elseif (im <= 8.2e+138)
		tmp = 1.0 / (re ^ 3.0);
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 800.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 8.2e+138], N[(1.0 / N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 800

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.7%

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

   if 800 < im < 8.19999999999999961e138

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{{\sin re}^{-3}} \]

   7. Taylor expanded in re around 0 38.7%

      \[\leadsto \color{blue}{\frac{1}{{re}^{3}}} \]

   if 8.19999999999999961e138 < 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. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. sub-neg 100.0%

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

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   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 97.2%

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

      1. +-commutative 97.2%

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

      2. unpow2 97.2%

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

      3. fma-def 97.2%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Simplified 97.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   7. Taylor expanded in re around 0 72.9%

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

      1. associate-*r* 72.9%

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

      2. *-commutative 72.9%

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

      3. +-commutative 72.9%

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

      4. unpow2 72.9%

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

      5. fma-def 72.9%

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

   9. Simplified 72.9%

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

   10. Taylor expanded in im around inf 72.9%

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

       1. associate-*r* 72.9%

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

   12. Simplified 72.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{{re}^{3}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]

Alternative 10: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125.0) (sin re) (* re (+ (* 0.5 (exp im)) 4.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = sin(re);
	} else {
		tmp = re * ((0.5 * exp(im)) + 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 125.0d0) then
        tmp = sin(re)
    else
        tmp = re * ((0.5d0 * exp(im)) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = Math.sin(re);
	} else {
		tmp = re * ((0.5 * Math.exp(im)) + 4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 125.0:
		tmp = math.sin(re)
	else:
		tmp = re * ((0.5 * math.exp(im)) + 4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 125.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(Float64(0.5 * exp(im)) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 125.0)
		tmp = sin(re);
	else
		tmp = re * ((0.5 * exp(im)) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 125.0], N[Sin[re], $MachinePrecision], N[(re * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 125

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.7%

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

   if 125 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around 0 77.2%

      \[\leadsto \color{blue}{re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot e^{im} + 4\right)\\ \end{array} \]

Alternative 11: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{{re}^{3}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 500.0)
   (sin re)
   (if (<= im 2.1e+138)
     (/ 1.0 (pow re 3.0))
     (* re (+ 4.5 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = sin(re);
	} else if (im <= 2.1e+138) {
		tmp = 1.0 / pow(re, 3.0);
	} else {
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 500.0d0) then
        tmp = sin(re)
    else if (im <= 2.1d+138) then
        tmp = 1.0d0 / (re ** 3.0d0)
    else
        tmp = re * (4.5d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.1e+138) {
		tmp = 1.0 / Math.pow(re, 3.0);
	} else {
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 500.0:
		tmp = math.sin(re)
	elif im <= 2.1e+138:
		tmp = 1.0 / math.pow(re, 3.0)
	else:
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 500.0)
		tmp = sin(re);
	elseif (im <= 2.1e+138)
		tmp = Float64(1.0 / (re ^ 3.0));
	else
		tmp = Float64(re * Float64(4.5 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 500.0)
		tmp = sin(re);
	elseif (im <= 2.1e+138)
		tmp = 1.0 / (re ^ 3.0);
	else
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 500.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.1e+138], N[(1.0 / N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(4.5 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 500

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.7%

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

   if 500 < im < 2.10000000000000007e138

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{{\sin re}^{-3}} \]

   7. Taylor expanded in re around 0 38.7%

      \[\leadsto \color{blue}{\frac{1}{{re}^{3}}} \]

   if 2.10000000000000007e138 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 97.2%

      \[\leadsto \color{blue}{0.25 \cdot \left({im}^{2} \cdot \sin re\right) + \left(0.5 \cdot \left(im \cdot \sin re\right) + 4.5 \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 97.2%

         \[\leadsto \color{blue}{\left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) + 4.5 \cdot \sin re} \]

      2. +-commutative 97.2%

         \[\leadsto \color{blue}{4.5 \cdot \sin re + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right)} \]

      3. *-commutative 97.2%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      4. associate-*r* 97.2%

         \[\leadsto \sin re \cdot 4.5 + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \sin re} + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      5. associate-*r* 97.2%

         \[\leadsto \sin re \cdot 4.5 + \left(\left(0.25 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.5 \cdot im\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 97.2%

         \[\leadsto \sin re \cdot 4.5 + \color{blue}{\sin re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      7. distribute-lft-out 97.2%

         \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      8. unpow2 97.2%

         \[\leadsto \sin re \cdot \left(4.5 + \left(0.25 \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot im\right)\right) \]

      9. associate-*r* 97.2%

         \[\leadsto \sin re \cdot \left(4.5 + \left(\color{blue}{\left(0.25 \cdot im\right) \cdot im} + 0.5 \cdot im\right)\right) \]

      10. distribute-rgt-out 97.2%

          \[\leadsto \sin re \cdot \left(4.5 + \color{blue}{im \cdot \left(0.25 \cdot im + 0.5\right)}\right) \]

   9. Simplified 97.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + im \cdot \left(0.25 \cdot im + 0.5\right)\right)} \]

   10. Taylor expanded in re around 0 72.9%

       \[\leadsto \color{blue}{re \cdot \left(4.5 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{{re}^{3}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 12: 61.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e+17) (sin re) (* re (+ 4.5 (* im (+ 0.5 (* im 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e+17) {
		tmp = sin(re);
	} else {
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	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+17) then
        tmp = sin(re)
    else
        tmp = re * (4.5d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9e+17) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9e+17:
		tmp = math.sin(re)
	else:
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e+17)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(4.5 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9e+17)
		tmp = sin(re);
	else
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e+17], N[Sin[re], $MachinePrecision], N[(re * N[(4.5 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9e17

   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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 68.0%

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

   if 9e17 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 59.9%

      \[\leadsto \color{blue}{0.25 \cdot \left({im}^{2} \cdot \sin re\right) + \left(0.5 \cdot \left(im \cdot \sin re\right) + 4.5 \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 59.9%

         \[\leadsto \color{blue}{\left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) + 4.5 \cdot \sin re} \]

      2. +-commutative 59.9%

         \[\leadsto \color{blue}{4.5 \cdot \sin re + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right)} \]

      3. *-commutative 59.9%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      4. associate-*r* 59.9%

         \[\leadsto \sin re \cdot 4.5 + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \sin re} + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      5. associate-*r* 59.9%

         \[\leadsto \sin re \cdot 4.5 + \left(\left(0.25 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.5 \cdot im\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 59.9%

         \[\leadsto \sin re \cdot 4.5 + \color{blue}{\sin re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      7. distribute-lft-out 59.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      8. unpow2 59.9%

         \[\leadsto \sin re \cdot \left(4.5 + \left(0.25 \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot im\right)\right) \]

      9. associate-*r* 59.9%

         \[\leadsto \sin re \cdot \left(4.5 + \left(\color{blue}{\left(0.25 \cdot im\right) \cdot im} + 0.5 \cdot im\right)\right) \]

      10. distribute-rgt-out 59.9%

          \[\leadsto \sin re \cdot \left(4.5 + \color{blue}{im \cdot \left(0.25 \cdot im + 0.5\right)}\right) \]

   9. Simplified 59.9%

      \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + im \cdot \left(0.25 \cdot im + 0.5\right)\right)} \]

   10. Taylor expanded in re around 0 53.8%

       \[\leadsto \color{blue}{re \cdot \left(4.5 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 13: 38.0% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125.0) re (* re (+ 4.5 (* im (+ 0.5 (* im 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = re;
	} else {
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 125.0d0) then
        tmp = re
    else
        tmp = re * (4.5d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = re;
	} else {
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 125.0:
		tmp = re
	else:
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 125.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(4.5 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 125.0)
		tmp = re;
	else
		tmp = re * (4.5 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 125.0], re, N[(re * N[(4.5 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 125

   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. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. sub-neg 100.0%

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

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   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 83.2%

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

      1. +-commutative 83.2%

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

      2. unpow2 83.2%

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

      3. fma-def 83.2%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Simplified 83.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   7. Taylor expanded in re around 0 48.3%

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

      1. associate-*r* 48.3%

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

      2. *-commutative 48.3%

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

      3. +-commutative 48.3%

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

      4. unpow2 48.3%

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

      5. fma-def 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in im around 0 34.6%

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

   if 125 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in im around 0 57.9%

      \[\leadsto \color{blue}{0.25 \cdot \left({im}^{2} \cdot \sin re\right) + \left(0.5 \cdot \left(im \cdot \sin re\right) + 4.5 \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 57.9%

         \[\leadsto \color{blue}{\left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) + 4.5 \cdot \sin re} \]

      2. +-commutative 57.9%

         \[\leadsto \color{blue}{4.5 \cdot \sin re + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right)} \]

      3. *-commutative 57.9%

         \[\leadsto \color{blue}{\sin re \cdot 4.5} + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      4. associate-*r* 57.9%

         \[\leadsto \sin re \cdot 4.5 + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \sin re} + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      5. associate-*r* 57.9%

         \[\leadsto \sin re \cdot 4.5 + \left(\left(0.25 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.5 \cdot im\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 57.9%

         \[\leadsto \sin re \cdot 4.5 + \color{blue}{\sin re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      7. distribute-lft-out 57.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      8. unpow2 57.9%

         \[\leadsto \sin re \cdot \left(4.5 + \left(0.25 \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot im\right)\right) \]

      9. associate-*r* 57.9%

         \[\leadsto \sin re \cdot \left(4.5 + \left(\color{blue}{\left(0.25 \cdot im\right) \cdot im} + 0.5 \cdot im\right)\right) \]

      10. distribute-rgt-out 57.9%

          \[\leadsto \sin re \cdot \left(4.5 + \color{blue}{im \cdot \left(0.25 \cdot im + 0.5\right)}\right) \]

   9. Simplified 57.9%

      \[\leadsto \color{blue}{\sin re \cdot \left(4.5 + im \cdot \left(0.25 \cdot im + 0.5\right)\right)} \]

   10. Taylor expanded in re around 0 52.0%

       \[\leadsto \color{blue}{re \cdot \left(4.5 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 14: 30.0% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + 0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 600.0) re (* re (+ 4.5 (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 600.0) {
		tmp = re;
	} else {
		tmp = re * (4.5 + (0.5 * 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 <= 600.0d0) then
        tmp = re
    else
        tmp = re * (4.5d0 + (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 600.0) {
		tmp = re;
	} else {
		tmp = re * (4.5 + (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 600.0:
		tmp = re
	else:
		tmp = re * (4.5 + (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 600.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(4.5 + Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 600.0)
		tmp = re;
	else
		tmp = re * (4.5 + (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 600.0], re, N[(re * N[(4.5 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 600

   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. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. sub-neg 100.0%

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

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   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 83.2%

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

      1. +-commutative 83.2%

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

      2. unpow2 83.2%

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

      3. fma-def 83.2%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Simplified 83.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   7. Taylor expanded in re around 0 48.3%

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

      1. associate-*r* 48.3%

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

      2. *-commutative 48.3%

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

      3. +-commutative 48.3%

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

      4. unpow2 48.3%

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

      5. fma-def 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in im around 0 34.6%

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

   if 600 < 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. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \color{blue}{8}\right) \]

   7. Taylor expanded in re around 0 77.2%

      \[\leadsto \color{blue}{re \cdot \left(4 + 0.5 \cdot e^{im}\right)} \]

   8. Taylor expanded in im around 0 20.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot re\right) + 4.5 \cdot re} \]
   9. Step-by-step derivation

      1. associate-*r* 20.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot im\right) \cdot re} + 4.5 \cdot re \]

      2. *-commutative 20.2%

         \[\leadsto \color{blue}{\left(im \cdot 0.5\right)} \cdot re + 4.5 \cdot re \]

      3. distribute-rgt-out 20.2%

         \[\leadsto \color{blue}{re \cdot \left(im \cdot 0.5 + 4.5\right)} \]

   10. Simplified 20.2%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot 0.5 + 4.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(4.5 + 0.5 \cdot im\right)\\ \end{array} \]

Alternative 15: 26.8% 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. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

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

   8. associate-*r* 100.0%

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

   9. distribute-rgt-out-- 100.0%

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

   10. sub-neg 100.0%

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

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

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 77.5%

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

   1. +-commutative 77.5%

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

   2. unpow2 77.5%

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

   3. fma-def 77.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

6. Simplified 77.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

7. Taylor expanded in re around 0 49.2%

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

   1. associate-*r* 49.2%

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

   2. *-commutative 49.2%

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

   3. +-commutative 49.2%

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

   4. unpow2 49.2%

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

   5. fma-def 49.2%

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

9. Simplified 49.2%

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

10. Taylor expanded in im around 0 27.6%

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

11. Final simplification 27.6%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023318 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))