Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 20.0s

Alternatives: 20

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 92.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   2. unpow2 92.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   3. unpow2 92.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   4. hypot-undefine 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

5. Final simplification 99.7%

   \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]
6. Add Preprocessing

Alternative 2: 44.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|t\_1\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.02)
     (* ky (fabs t_1))
     (if (<= (sin kx) 1e-94) (sin th) (* (sin ky) t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = ky * fabs(t_1);
	} else if (sin(kx) <= 1e-94) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    if (sin(kx) <= (-0.02d0)) then
        tmp = ky * abs(t_1)
    else if (sin(kx) <= 1d-94) then
        tmp = sin(th)
    else
        tmp = sin(ky) * t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = ky * Math.abs(t_1);
	} else if (Math.sin(kx) <= 1e-94) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = ky * math.fabs(t_1)
	elif math.sin(kx) <= 1e-94:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = Float64(ky * abs(t_1));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = ky * abs(t_1);
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = sin(ky) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-94], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 14.2%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 14.2%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 14.2%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 10.3%

         \[\leadsto ky \cdot \color{blue}{\left(\sqrt{\frac{\sin th}{\sin kx}} \cdot \sqrt{\frac{\sin th}{\sin kx}}\right)} \]

      2. sqrt-unprod 37.1%

         \[\leadsto ky \cdot \color{blue}{\sqrt{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

      3. pow2 37.1%

         \[\leadsto ky \cdot \sqrt{\color{blue}{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

          \[\leadsto ky \cdot \sqrt{\color{blue}{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

       2. rem-sqrt-square 38.6%

          \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   11. Simplified 38.6%

       \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   if -0.0200000000000000004 < (sin.f64 kx) < 9.9999999999999996e-95

   1. Initial program 85.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 85.2%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 38.8%

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

   if 9.9999999999999996e-95 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 59.9%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (* ky (fabs (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-94) (sin th) (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = ky * fabs((sin(th) / sin(kx)));
	} else if (sin(kx) <= 1e-94) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = ky * abs((sin(th) / sin(kx)))
    else if (sin(kx) <= 1d-94) then
        tmp = sin(th)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = ky * Math.abs((Math.sin(th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-94) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = ky * math.fabs((math.sin(th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-94:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = Float64(ky * abs(Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = ky * abs((sin(th) / sin(kx)));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-94], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 14.2%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 14.2%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 14.2%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 10.3%

         \[\leadsto ky \cdot \color{blue}{\left(\sqrt{\frac{\sin th}{\sin kx}} \cdot \sqrt{\frac{\sin th}{\sin kx}}\right)} \]

      2. sqrt-unprod 37.1%

         \[\leadsto ky \cdot \color{blue}{\sqrt{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

      3. pow2 37.1%

         \[\leadsto ky \cdot \sqrt{\color{blue}{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

          \[\leadsto ky \cdot \sqrt{\color{blue}{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

       2. rem-sqrt-square 38.6%

          \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   11. Simplified 38.6%

       \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   if -0.0200000000000000004 < (sin.f64 kx) < 9.9999999999999996e-95

   1. Initial program 85.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 85.2%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 38.8%

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

   if 9.9999999999999996e-95 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 59.9%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. clear-num 59.9%

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\sin kx}{\sin th}}} \]

      2. un-div-inv 59.9%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sin kx}{\sin th}}} \]

   7. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sin kx}{\sin th}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (* ky (fabs (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-94) (sin th) (/ (sin th) (/ (sin kx) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = ky * fabs((sin(th) / sin(kx)));
	} else if (sin(kx) <= 1e-94) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = ky * abs((sin(th) / sin(kx)))
    else if (sin(kx) <= 1d-94) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = ky * Math.abs((Math.sin(th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-94) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = ky * math.fabs((math.sin(th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-94:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = Float64(ky * abs(Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = ky * abs((sin(th) / sin(kx)));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-94], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 14.2%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 14.2%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 14.2%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 10.3%

         \[\leadsto ky \cdot \color{blue}{\left(\sqrt{\frac{\sin th}{\sin kx}} \cdot \sqrt{\frac{\sin th}{\sin kx}}\right)} \]

      2. sqrt-unprod 37.1%

         \[\leadsto ky \cdot \color{blue}{\sqrt{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

      3. pow2 37.1%

         \[\leadsto ky \cdot \sqrt{\color{blue}{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

          \[\leadsto ky \cdot \sqrt{\color{blue}{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

       2. rem-sqrt-square 38.6%

          \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   11. Simplified 38.6%

       \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   if -0.0200000000000000004 < (sin.f64 kx) < 9.9999999999999996e-95

   1. Initial program 85.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 85.2%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 38.8%

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

   if 9.9999999999999996e-95 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-undefine 99.5%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0 59.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative 59.9%

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sin kx}} \]

      2. clear-num 59.9%

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sin kx}{\sin ky}}} \]

      3. un-div-inv 60.0%

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sin kx}{\sin ky}}} \]

   7. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sin kx}{\sin ky}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (fabs (* (sin th) (/ (sin ky) (sin kx))))
   (if (<= (sin kx) 1e-94) (sin th) (/ (sin th) (/ (sin kx) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else if (sin(kx) <= 1e-94) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = abs((sin(th) * (sin(ky) / sin(kx))))
    else if (sin(kx) <= 1d-94) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 1e-94) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	elif math.sin(kx) <= 1e-94:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-94], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-undefine 99.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0 13.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. clear-num 13.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin ky}}} \cdot \sin th \]

      2. associate-/r/ 13.7%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin kx} \cdot \sin ky\right)} \cdot \sin th \]

   7. Applied egg-rr 13.7%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin kx} \cdot \sin ky\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 11.8%

         \[\leadsto \color{blue}{\sqrt{\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th} \cdot \sqrt{\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th}} \]

      2. sqrt-unprod 24.8%

         \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\right) \cdot \left(\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\right)}} \]

      3. pow2 24.8%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\right)}^{2}}} \]

      4. *-commutative 24.8%

         \[\leadsto \sqrt{{\color{blue}{\left(\sin th \cdot \left(\frac{1}{\sin kx} \cdot \sin ky\right)\right)}}^{2}} \]

      5. associate-*l/ 24.8%

         \[\leadsto \sqrt{{\left(\sin th \cdot \color{blue}{\frac{1 \cdot \sin ky}{\sin kx}}\right)}^{2}} \]

      6. *-un-lft-identity 24.8%

         \[\leadsto \sqrt{{\left(\sin th \cdot \frac{\color{blue}{\sin ky}}{\sin kx}\right)}^{2}} \]

   9. Applied egg-rr 24.8%

      \[\leadsto \color{blue}{\sqrt{{\left(\sin th \cdot \frac{\sin ky}{\sin kx}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 24.8%

          \[\leadsto \sqrt{\color{blue}{\left(\sin th \cdot \frac{\sin ky}{\sin kx}\right) \cdot \left(\sin th \cdot \frac{\sin ky}{\sin kx}\right)}} \]

       2. rem-sqrt-square 45.1%

          \[\leadsto \color{blue}{\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|} \]

   11. Simplified 45.1%

       \[\leadsto \color{blue}{\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|} \]

   if -0.0200000000000000004 < (sin.f64 kx) < 9.9999999999999996e-95

   1. Initial program 85.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 85.2%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 38.8%

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

   if 9.9999999999999996e-95 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-undefine 99.5%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0 59.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative 59.9%

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sin kx}} \]

      2. clear-num 59.9%

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sin kx}{\sin ky}}} \]

      3. un-div-inv 60.0%

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sin kx}{\sin ky}}} \]

   7. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sin kx}{\sin ky}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (sin ky) (/ (sin th) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	return sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), Math.sin(kx)));
}

Python

def code(kx, ky, th):
	return math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), math.sin(kx)))

Julia

function code(kx, ky, th)
	return Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), sin(kx))))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-*l/ 89.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. associate-/l* 92.9%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. unpow2 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

   4. sqr-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

   5. sin-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

   6. sin-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   7. unpow2 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

   8. unpow2 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   9. sin-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

   10. sin-neg 92.9%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

   11. sqr-neg 92.9%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

   12. unpow2 92.9%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Final simplification 99.6%

   \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
6. Add Preprocessing

Alternative 7: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.075:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.075)
   (/
    (sin ky)
    (* (hypot (sin kx) (sin ky)) (+ (/ 1.0 th) (* th 0.16666666666666666))))
   (/ (* ky (sin th)) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.075) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else {
		tmp = (ky * sin(th)) / hypot(sin(ky), sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.075) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else {
		tmp = (ky * Math.sin(th)) / Math.hypot(Math.sin(ky), Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.075:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)))
	else:
		tmp = (ky * math.sin(th)) / math.hypot(math.sin(ky), math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.075)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	else
		tmp = Float64(Float64(ky * sin(th)) / hypot(sin(ky), sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.075)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	else
		tmp = (ky * sin(th)) / hypot(sin(ky), sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.075], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.0749999999999999972

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 89.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 93.8%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 93.8%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 93.8%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 93.8%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.5%

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      2. un-div-inv 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   7. Taylor expanded in th around 0 62.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   8. Step-by-step derivation

      1. +-commutative 62.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      2. +-commutative 62.7%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      3. unpow2 62.7%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      4. unpow2 62.7%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      5. hypot-undefine 65.6%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. associate-*r* 65.6%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. +-commutative 65.6%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right) + \left(0.16666666666666666 \cdot th\right) \cdot \sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      8. unpow2 65.6%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right) + \left(0.16666666666666666 \cdot th\right) \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      9. unpow2 65.6%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right) + \left(0.16666666666666666 \cdot th\right) \cdot \sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      10. hypot-undefine 65.7%

          \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right) + \left(0.16666666666666666 \cdot th\right) \cdot \color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      11. distribute-rgt-out 65.7%

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]

   9. Simplified 65.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]

   if 0.0749999999999999972 < th

   1. Initial program 89.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 89.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 89.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 89.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 89.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 89.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   7. Taylor expanded in ky around 0 57.4%

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.075:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (* ky (fabs (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-94) (sin th) (/ (sin th) (/ (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = ky * fabs((sin(th) / sin(kx)));
	} else if (sin(kx) <= 1e-94) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = ky * abs((sin(th) / sin(kx)))
    else if (sin(kx) <= 1d-94) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = ky * Math.abs((Math.sin(th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-94) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = ky * math.fabs((math.sin(th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-94:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = Float64(ky * abs(Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = ky * abs((sin(th) / sin(kx)));
	elseif (sin(kx) <= 1e-94)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-94], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 14.2%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 14.2%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 14.2%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 10.3%

         \[\leadsto ky \cdot \color{blue}{\left(\sqrt{\frac{\sin th}{\sin kx}} \cdot \sqrt{\frac{\sin th}{\sin kx}}\right)} \]

      2. sqrt-unprod 37.1%

         \[\leadsto ky \cdot \color{blue}{\sqrt{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

      3. pow2 37.1%

         \[\leadsto ky \cdot \sqrt{\color{blue}{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin kx}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

          \[\leadsto ky \cdot \sqrt{\color{blue}{\frac{\sin th}{\sin kx} \cdot \frac{\sin th}{\sin kx}}} \]

       2. rem-sqrt-square 38.6%

          \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   11. Simplified 38.6%

       \[\leadsto ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]

   if -0.0200000000000000004 < (sin.f64 kx) < 9.9999999999999996e-95

   1. Initial program 85.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 85.2%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 85.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 85.2%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 38.8%

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

   if 9.9999999999999996e-95 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      2. hypot-undefine 98.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      3. unpow2 98.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      4. unpow2 98.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      5. +-commutative 98.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      7. *-commutative 99.5%

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. clear-num 99.5%

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      9. un-div-inv 99.6%

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      10. +-commutative 99.6%

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. unpow2 99.6%

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      12. unpow2 99.6%

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      13. hypot-undefine 99.5%

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   7. Taylor expanded in ky around 0 53.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.5:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 1.5)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (* (sin th) (fabs (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.5) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.5) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 1.5:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 1.5)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 1.5)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 1.5], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.5

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 89.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 93.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 93.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 93.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 93.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   7. Taylor expanded in th around 0 58.5%

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   8. Step-by-step derivation

      1. *-commutative 58.5%

         \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. associate-/l* 64.4%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   9. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   if 1.5 < th

   1. Initial program 89.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 89.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 89.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 89.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-undefine 99.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0 25.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 17.8%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{\sin ky}{\sin kx}} \cdot \sqrt{\frac{\sin ky}{\sin kx}}\right)} \cdot \sin th \]

      2. sqrt-unprod 18.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      3. pow2 18.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. unpow2 18.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      2. rem-sqrt-square 34.3%

         \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]

   9. Simplified 34.3%

      \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.5:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.5:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 1.5)
   (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
   (* (sin th) (fabs (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.5) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.5) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 1.5:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 1.5)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 1.5)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 1.5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.5

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 89.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 93.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 93.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 93.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 93.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 93.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   7. Taylor expanded in th around 0 58.5%

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   8. Step-by-step derivation

      1. associate-/l* 64.4%

         \[\leadsto \color{blue}{th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      2. *-commutative 64.4%

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th} \]

   9. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th} \]

   if 1.5 < th

   1. Initial program 89.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 89.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 89.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 89.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-undefine 99.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0 25.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 17.8%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{\sin ky}{\sin kx}} \cdot \sqrt{\frac{\sin ky}{\sin kx}}\right)} \cdot \sin th \]

      2. sqrt-unprod 18.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      3. pow2 18.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. unpow2 18.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      2. rem-sqrt-square 34.3%

         \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]

   9. Simplified 34.3%

      \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.5:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= th 0.0005) (* (/ (sin ky) t_1) th) (/ (* ky (sin th)) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (th <= 0.0005) {
		tmp = (sin(ky) / t_1) * th;
	} else {
		tmp = (ky * sin(th)) / t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (th <= 0.0005) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else {
		tmp = (ky * Math.sin(th)) / t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if th <= 0.0005:
		tmp = (math.sin(ky) / t_1) * th
	else:
		tmp = (ky * math.sin(th)) / t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (th <= 0.0005)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	else
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (th <= 0.0005)
		tmp = (sin(ky) / t_1) * th;
	else
		tmp = (ky * sin(th)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, 0.0005], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 5.0000000000000001e-4

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 89.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 93.8%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 93.8%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 93.8%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 93.8%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 93.8%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   7. Taylor expanded in th around 0 58.7%

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   8. Step-by-step derivation

      1. associate-/l* 64.6%

         \[\leadsto \color{blue}{th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      2. *-commutative 64.6%

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th} \]

   9. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th} \]

   if 5.0000000000000001e-4 < th

   1. Initial program 89.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 89.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 89.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 89.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 89.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 89.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 89.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   7. Taylor expanded in ky around 0 57.4%

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.55 \cdot 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.55e-94) (sin th) (* (sin th) (fabs (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.55e-94) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 2.55d-94) then
        tmp = sin(th)
    else
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.55e-94) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 2.55e-94:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.55e-94)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 2.55e-94)
		tmp = sin(th);
	else
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.55e-94], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.5500000000000001e-94

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 85.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 90.0%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 90.0%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 90.0%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 90.0%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 27.8%

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

   if 2.5500000000000001e-94 < kx

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-undefine 99.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0 41.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 27.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{\sin ky}{\sin kx}} \cdot \sqrt{\frac{\sin ky}{\sin kx}}\right)} \cdot \sin th \]

      2. sqrt-unprod 33.1%

         \[\leadsto \color{blue}{\sqrt{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      3. pow2 33.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]

   7. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. unpow2 33.1%

         \[\leadsto \sqrt{\color{blue}{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      2. rem-sqrt-square 37.9%

         \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]

   9. Simplified 37.9%

      \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.55 \cdot 10^{-94}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-116}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-116) (* ky (/ (sin th) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-116) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-116) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-116) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-116:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-116)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-116)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-116], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2e-116

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 86.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 89.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 32.8%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 34.4%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 34.4%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   if 2e-116 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 96.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 59.7%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-116}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-116) (/ ky (/ (sin kx) (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-116) {
		tmp = ky / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-116) then
        tmp = ky / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-116) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-116:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-116)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-116)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-116], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2e-116

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 86.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 89.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 32.8%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 34.4%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 34.4%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]
   8. Step-by-step derivation

      1. clear-num 34.4%

         \[\leadsto ky \cdot \color{blue}{\frac{1}{\frac{\sin kx}{\sin th}}} \]

      2. un-div-inv 34.4%

         \[\leadsto \color{blue}{\frac{ky}{\frac{\sin kx}{\sin th}}} \]

   9. Applied egg-rr 34.4%

      \[\leadsto \color{blue}{\frac{ky}{\frac{\sin kx}{\sin th}}} \]

   if 2e-116 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 96.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 59.7%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-116) (/ (sin th) (/ (sin kx) ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-116) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-116) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-116) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-116:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-116)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-116)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-116], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2e-116

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 86.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 89.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 89.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 89.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 94.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      2. hypot-undefine 86.6%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      3. unpow2 86.6%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      4. unpow2 86.6%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      5. +-commutative 86.6%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. associate-*l/ 89.9%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      7. *-commutative 89.9%

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. clear-num 89.9%

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      9. un-div-inv 90.0%

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      10. +-commutative 90.0%

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. unpow2 90.0%

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      12. unpow2 90.0%

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      13. hypot-undefine 99.7%

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   7. Taylor expanded in ky around 0 34.4%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 2e-116 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 96.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 59.7%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.9 \cdot 10^{-184}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.9e-184) (* ky (/ th (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.9e-184) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.9d-184) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.9e-184) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.9e-184:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.9e-184)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.9e-184)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.9e-184], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.90000000000000008e-184

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 88.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 90.1%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 90.1%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 90.1%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 90.1%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 34.7%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 36.4%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 36.4%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in th around 0 19.1%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 20.7%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 20.7%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   if 1.90000000000000008e-184 < ky

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 91.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 96.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 96.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 96.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 96.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 39.1%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.9 \cdot 10^{-184}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.3e-114) (* ky (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.3e-114) {
		tmp = ky * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 4.3d-114) then
        tmp = ky * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.3e-114) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.3e-114:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.3e-114)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.3e-114)
		tmp = ky * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.3e-114], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.3e-114

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 85.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 89.3%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 89.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 89.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 89.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 89.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 34.4%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 36.1%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 36.1%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in kx around 0 23.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 25.1%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   10. Simplified 25.1%

       \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   if 4.3e-114 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 96.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 38.5%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 24.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.1 \cdot 10^{-184}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.1e-184) (* ky (/ th kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.1e-184) {
		tmp = ky * (th / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 2.1d-184) then
        tmp = ky * (th / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.1e-184) {
		tmp = ky * (th / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.1e-184:
		tmp = ky * (th / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.1e-184)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.1e-184)
		tmp = ky * (th / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.1e-184], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.0999999999999999e-184

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 88.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 90.1%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 90.1%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 90.1%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 90.1%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 90.1%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 34.7%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 36.4%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 36.4%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in th around 0 19.1%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 20.7%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 20.7%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   11. Taylor expanded in kx around 0 18.8%

       \[\leadsto \color{blue}{\frac{ky \cdot th}{kx}} \]
   12. Step-by-step derivation

       1. associate-/l* 20.5%

          \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   13. Simplified 20.5%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   if 2.0999999999999999e-184 < ky

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 91.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 96.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 96.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 96.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 96.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 96.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 39.1%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.1 \cdot 10^{-184}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 18.2% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 8.5 \cdot 10^{-174}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 8.5e-174) (* ky (/ th kx)) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8.5e-174) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 8.5d-174) then
        tmp = ky * (th / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8.5e-174) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 8.5e-174:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 8.5e-174)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 8.5e-174)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 8.5e-174], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 8.4999999999999996e-174

   1. Initial program 88.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 86.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 88.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 88.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 88.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 88.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 88.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 34.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 36.1%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 36.1%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in th around 0 19.3%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 20.8%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 20.8%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   11. Taylor expanded in kx around 0 19.0%

       \[\leadsto \color{blue}{\frac{ky \cdot th}{kx}} \]
   12. Step-by-step derivation

       1. associate-/l* 20.7%

          \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   13. Simplified 20.7%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   if 8.4999999999999996e-174 < ky

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 94.0%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* 99.3%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. unpow2 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      4. sqr-neg 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      5. sin-neg 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

      6. sin-neg 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      7. unpow2 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

      9. sin-neg 99.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

      10. sin-neg 99.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

      11. sqr-neg 99.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      12. unpow2 99.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 94.0%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied egg-rr 94.0%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   7. Taylor expanded in th around 0 50.5%

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   8. Taylor expanded in kx around 0 22.5%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 8.5 \cdot 10^{-174}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 14.0% accurate, 709.0× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-*l/ 89.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. associate-/l* 92.9%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. unpow2 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

   4. sqr-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \]

   5. sin-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \]

   6. sin-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   7. unpow2 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \]

   8. unpow2 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   9. sin-neg 92.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right) + {\sin ky}^{2}}} \]

   10. sin-neg 92.9%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)} + {\sin ky}^{2}}} \]

   11. sqr-neg 92.9%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

   12. unpow2 92.9%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 95.1%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

6. Applied egg-rr 95.1%

   \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

7. Taylor expanded in th around 0 46.1%

   \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

8. Taylor expanded in kx around 0 14.0%

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

9. Final simplification 14.0%

   \[\leadsto th \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))