Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.5% → 99.7%

Time: 28.3s

Alternatives: 23

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: 94.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.0%

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

   1. +-commutative 92.0%

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

   2. unpow2 92.0%

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

   3. unpow2 92.0%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 53.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (fabs (/ (sin ky) (sin kx))))))
   (if (<= (sin ky) -0.01)
     (fabs (sin th))
     (if (<= (sin ky) -5e-78)
       (/ (* th ky) (hypot (sin ky) (sin kx)))
       (if (<= (sin ky) -2e-101)
         (/ 1.0 (sqrt (pow (sin th) -2.0)))
         (if (<= (sin ky) -5e-306)
           (fabs (* (sin ky) (/ (sin th) (sin kx))))
           (if (<= (sin ky) 2e-115)
             t_1
             (if (<= (sin ky) 2e-55)
               (/ (sin th) (+ 1.0 (/ (* 0.5 (* kx kx)) (pow (sin ky) 2.0))))
               (if (<= (sin ky) 1e-22) t_1 (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * fabs((sin(ky) / sin(kx)));
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-78) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= -2e-101) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= -5e-306) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(ky) <= 2e-115) {
		tmp = t_1;
	} else if (sin(ky) <= 2e-55) {
		tmp = sin(th) / (1.0 + ((0.5 * (kx * kx)) / pow(sin(ky), 2.0)));
	} else if (sin(ky) <= 1e-22) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-78) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= -2e-101) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= -5e-306) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(ky) <= 2e-115) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 2e-55) {
		tmp = Math.sin(th) / (1.0 + ((0.5 * (kx * kx)) / Math.pow(Math.sin(ky), 2.0)));
	} else if (Math.sin(ky) <= 1e-22) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-78:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= -2e-101:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= -5e-306:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(ky) <= 2e-115:
		tmp = t_1
	elif math.sin(ky) <= 2e-55:
		tmp = math.sin(th) / (1.0 + ((0.5 * (kx * kx)) / math.pow(math.sin(ky), 2.0)))
	elif math.sin(ky) <= 1e-22:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))))
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-78)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= -2e-101)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= -5e-306)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		tmp = t_1;
	elseif (sin(ky) <= 2e-55)
		tmp = Float64(sin(th) / Float64(1.0 + Float64(Float64(0.5 * Float64(kx * kx)) / (sin(ky) ^ 2.0))));
	elseif (sin(ky) <= 1e-22)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) * abs((sin(ky) / sin(kx)));
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-78)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= -2e-101)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= -5e-306)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		tmp = t_1;
	elseif (sin(ky) <= 2e-55)
		tmp = sin(th) / (1.0 + ((0.5 * (kx * kx)) / (sin(ky) ^ 2.0)));
	elseif (sin(ky) <= 1e-22)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-78], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-101], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-306], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-115], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-55], N[(N[Sin[th], $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-22], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.1%

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

      3. sqrt-unprod 25.8%

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

      4. pow2 25.8%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 25.8%

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

      2. rem-sqrt-square 33.9%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 33.9%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.0100000000000000002 < (sin.f64 ky) < -4.9999999999999996e-78

   1. Initial program 99.8%

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

      1. associate-*l/ 95.3%

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

      2. +-commutative 95.3%

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

      3. unpow2 95.3%

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

      4. unpow2 95.3%

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

      5. hypot-def 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in th around 0 53.2%

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

   5. Taylor expanded in ky around 0 53.1%

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

   if -4.9999999999999996e-78 < (sin.f64 ky) < -2.0000000000000001e-101

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in kx around 0 2.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 71.8%

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

      3. inv-pow 71.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 71.8%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 71.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 71.8%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   8. Applied egg-rr 71.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -2.0000000000000001e-101 < (sin.f64 ky) < -4.99999999999999998e-306

   1. Initial program 76.5%

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

      1. +-commutative 76.5%

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

      2. unpow2 76.5%

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

      3. unpow2 76.5%

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

      4. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in ky around 0 57.9%

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

      1. add-sqr-sqrt 47.1%

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

      2. sqrt-unprod 50.9%

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

      3. pow2 50.9%

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

      4. *-commutative 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. rem-sqrt-square 67.1%

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

      3. associate-*r/ 61.6%

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

      4. associate-/l* 67.3%

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

      5. associate-/r/ 67.3%

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

   8. Simplified 67.3%

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

   if -4.99999999999999998e-306 < (sin.f64 ky) < 2.0000000000000001e-115 or 1.99999999999999999e-55 < (sin.f64 ky) < 1e-22

   1. Initial program 76.5%

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

      1. +-commutative 76.5%

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

      2. unpow2 76.5%

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

      3. unpow2 76.5%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 59.9%

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

      1. add-sqr-sqrt 46.0%

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

      2. sqrt-unprod 57.3%

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

      3. pow2 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. unpow2 57.3%

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

      2. rem-sqrt-square 77.5%

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

   8. Simplified 77.5%

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

   if 2.0000000000000001e-115 < (sin.f64 ky) < 1.99999999999999999e-55

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. unpow2 100.0%

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

      4. hypot-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 78.7%

      \[\leadsto \frac{\sin th}{\color{blue}{1 + 0.5 \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 78.7%

         \[\leadsto \frac{\sin th}{1 + \color{blue}{\frac{0.5 \cdot {kx}^{2}}{{\sin ky}^{2}}}} \]

      2. unpow2 78.7%

         \[\leadsto \frac{\sin th}{1 + \frac{0.5 \cdot \color{blue}{\left(kx \cdot kx\right)}}{{\sin ky}^{2}}} \]

   8. Simplified 78.7%

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}} \]

   if 1e-22 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 65.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 7 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-22}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 48.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-260}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (fabs (sin th))
   (if (<= (sin ky) -5e-78)
     (/ (* th ky) (hypot (sin ky) (sin kx)))
     (if (<= (sin ky) -2e-101)
       (/ 1.0 (sqrt (pow (sin th) -2.0)))
       (if (<= (sin ky) -1e-260)
         (fabs (* (sin ky) (/ (sin th) (sin kx))))
         (if (<= (sin ky) 2e-115) (/ (sin th) (/ (sin kx) ky)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-78) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= -2e-101) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= -1e-260) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(ky) <= 2e-115) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-78) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= -2e-101) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= -1e-260) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(ky) <= 2e-115) {
		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) <= -0.01:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-78:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= -2e-101:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= -1e-260:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(ky) <= 2e-115:
		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) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-78)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= -2e-101)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= -1e-260)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		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) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-78)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= -2e-101)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= -1e-260)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		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], -0.01], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-78], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-101], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-260], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-115], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.1%

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

      3. sqrt-unprod 25.8%

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

      4. pow2 25.8%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 25.8%

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

      2. rem-sqrt-square 33.9%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 33.9%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.0100000000000000002 < (sin.f64 ky) < -4.9999999999999996e-78

   1. Initial program 99.8%

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

      1. associate-*l/ 95.3%

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

      2. +-commutative 95.3%

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

      3. unpow2 95.3%

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

      4. unpow2 95.3%

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

      5. hypot-def 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in th around 0 53.2%

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

   5. Taylor expanded in ky around 0 53.1%

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

   if -4.9999999999999996e-78 < (sin.f64 ky) < -2.0000000000000001e-101

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in kx around 0 2.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 71.8%

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

      3. inv-pow 71.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 71.8%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 71.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 71.8%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   8. Applied egg-rr 71.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -2.0000000000000001e-101 < (sin.f64 ky) < -9.99999999999999961e-261

   1. Initial program 82.6%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. unpow2 82.6%

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

      4. hypot-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in ky around 0 54.4%

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

      1. add-sqr-sqrt 47.0%

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

      2. sqrt-unprod 54.3%

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

      3. pow2 54.3%

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

      4. *-commutative 54.3%

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

   6. Applied egg-rr 54.3%

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

      1. unpow2 54.3%

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

      2. rem-sqrt-square 72.5%

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

      3. associate-*r/ 66.1%

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

      4. associate-/l* 72.8%

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

      5. associate-/r/ 72.8%

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

   8. Simplified 72.8%

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

   if -9.99999999999999961e-261 < (sin.f64 ky) < 2.0000000000000001e-115

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. unpow2 68.0%

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

      3. unpow2 68.0%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 58.0%

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

      1. associate-/l* 62.5%

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

   6. Simplified 62.5%

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

   if 2.0000000000000001e-115 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 62.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 6 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-260}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 48.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (fabs (sin th))
   (if (<= (sin ky) -5e-78)
     (/ (* th ky) (hypot (sin ky) (sin kx)))
     (if (<= (sin ky) -2e-101)
       (/ 1.0 (sqrt (pow (sin th) -2.0)))
       (if (<= (sin ky) -4e-220)
         (fabs (* (sin th) (/ ky (sin kx))))
         (if (<= (sin ky) 2e-115) (/ (sin th) (/ (sin kx) ky)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-78) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= -2e-101) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= -4e-220) {
		tmp = fabs((sin(th) * (ky / sin(kx))));
	} else if (sin(ky) <= 2e-115) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-78) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= -2e-101) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= -4e-220) {
		tmp = Math.abs((Math.sin(th) * (ky / Math.sin(kx))));
	} else if (Math.sin(ky) <= 2e-115) {
		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) <= -0.01:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-78:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= -2e-101:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= -4e-220:
		tmp = math.fabs((math.sin(th) * (ky / math.sin(kx))))
	elif math.sin(ky) <= 2e-115:
		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) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-78)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= -2e-101)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= -4e-220)
		tmp = abs(Float64(sin(th) * Float64(ky / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		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) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-78)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= -2e-101)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= -4e-220)
		tmp = abs((sin(th) * (ky / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		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], -0.01], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-78], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-101], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -4e-220], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-115], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.1%

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

      3. sqrt-unprod 25.8%

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

      4. pow2 25.8%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 25.8%

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

      2. rem-sqrt-square 33.9%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 33.9%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.0100000000000000002 < (sin.f64 ky) < -4.9999999999999996e-78

   1. Initial program 99.8%

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

      1. associate-*l/ 95.3%

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

      2. +-commutative 95.3%

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

      3. unpow2 95.3%

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

      4. unpow2 95.3%

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

      5. hypot-def 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in th around 0 53.2%

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

   5. Taylor expanded in ky around 0 53.1%

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

   if -4.9999999999999996e-78 < (sin.f64 ky) < -2.0000000000000001e-101

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in kx around 0 2.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 71.8%

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

      3. inv-pow 71.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 71.8%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 71.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 71.8%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   8. Applied egg-rr 71.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -2.0000000000000001e-101 < (sin.f64 ky) < -3.99999999999999997e-220

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. unpow2 86.9%

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

      3. unpow2 86.9%

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

      4. hypot-def 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*l/ 94.8%

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

      2. clear-num 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in ky around 0 48.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{\sin th \cdot ky}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 40.1%

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

      2. sqrt-unprod 53.1%

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

      3. pow2 53.1%

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

      4. clear-num 53.1%

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

      5. *-un-lft-identity 53.1%

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

      6. times-frac 52.9%

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

      7. /-rgt-identity 52.9%

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

   8. Applied egg-rr 52.9%

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

      1. unpow2 52.9%

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

      2. rem-sqrt-square 71.3%

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

   10. Simplified 71.3%

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

   if -3.99999999999999997e-220 < (sin.f64 ky) < 2.0000000000000001e-115

   1. Initial program 67.3%

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

      1. +-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. unpow2 67.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 58.3%

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

      1. associate-/l* 64.1%

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

   6. Simplified 64.1%

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

   if 2.0000000000000001e-115 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 62.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 6 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 76.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{th}{\frac{t_1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.01)
     (/ th (/ t_1 (sin ky)))
     (if (<= (sin ky) 0.002)
       (/ (sin th) (* t_1 (+ (* ky 0.16666666666666666) (/ 1.0 ky))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = th / (t_1 / sin(ky));
	} else if (sin(ky) <= 0.002) {
		tmp = sin(th) / (t_1 * ((ky * 0.16666666666666666) + (1.0 / ky)));
	} else {
		tmp = sin(th);
	}
	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 (Math.sin(ky) <= -0.01) {
		tmp = th / (t_1 / Math.sin(ky));
	} else if (Math.sin(ky) <= 0.002) {
		tmp = Math.sin(th) / (t_1 * ((ky * 0.16666666666666666) + (1.0 / ky)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = th / (t_1 / math.sin(ky))
	elif math.sin(ky) <= 0.002:
		tmp = math.sin(th) / (t_1 * ((ky * 0.16666666666666666) + (1.0 / ky)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(th / Float64(t_1 / sin(ky)));
	elseif (sin(ky) <= 0.002)
		tmp = Float64(sin(th) / Float64(t_1 * Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = th / (t_1 / sin(ky));
	elseif (sin(ky) <= 0.002)
		tmp = sin(th) / (t_1 * ((ky * 0.16666666666666666) + (1.0 / ky)));
	else
		tmp = sin(th);
	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[N[Sin[ky], $MachinePrecision], -0.01], N[(th / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.002], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 * N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 57.1%

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

      1. expm1-log1p-u 56.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 4.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 57.2%

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

      3. *-commutative 57.2%

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

      4. associate-*l/ 57.1%

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

      5. associate-*r/ 57.1%

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

   8. Simplified 57.1%

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

      1. associate-*r/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-/l* 57.2%

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

   10. Applied egg-rr 57.2%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 2e-3

   1. Initial program 84.1%

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

      1. +-commutative 84.1%

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

      2. unpow2 84.1%

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

      3. unpow2 84.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in ky around 0 99.4%

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

   if 2e-3 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 64.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 58.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{\left|\sin th \cdot ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (fabs (* (sin ky) (/ (sin th) (sin kx))))
   (if (<= (sin th) 5e-23)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (if (<= (sin th) 0.985)
       (sin th)
       (/ 1.0 (/ (sin kx) (fabs (* (sin th) ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.05) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(th) <= 5e-23) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(th) <= 0.985) {
		tmp = sin(th);
	} else {
		tmp = 1.0 / (sin(kx) / fabs((sin(th) * ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.05) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(th) <= 5e-23) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(th) <= 0.985) {
		tmp = Math.sin(th);
	} else {
		tmp = 1.0 / (Math.sin(kx) / Math.abs((Math.sin(th) * ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.05:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(th) <= 5e-23:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(th) <= 0.985:
		tmp = math.sin(th)
	else:
		tmp = 1.0 / (math.sin(kx) / math.fabs((math.sin(th) * ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.05)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(th) <= 5e-23)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(th) <= 0.985)
		tmp = sin(th);
	else
		tmp = Float64(1.0 / Float64(sin(kx) / abs(Float64(sin(th) * ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.05)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(th) <= 5e-23)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(th) <= 0.985)
		tmp = sin(th);
	else
		tmp = 1.0 / (sin(kx) / abs((sin(th) * ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 5e-23], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.985], N[Sin[th], $MachinePrecision], N[(1.0 / N[(N[Sin[kx], $MachinePrecision] / N[Abs[N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.050000000000000003

   1. Initial program 90.1%

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

      1. +-commutative 90.1%

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

      2. unpow2 90.1%

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

      3. unpow2 90.1%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 25.4%

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

      1. add-sqr-sqrt 15.7%

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

      2. sqrt-unprod 24.7%

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

      3. pow2 24.7%

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

      4. *-commutative 24.7%

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

   6. Applied egg-rr 24.7%

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

      1. unpow2 24.7%

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

      2. rem-sqrt-square 29.9%

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

      3. associate-*r/ 30.0%

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

      4. associate-/l* 30.0%

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

      5. associate-/r/ 30.0%

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

   8. Simplified 30.0%

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

   if -0.050000000000000003 < (sin.f64 th) < 5.0000000000000002e-23

   1. Initial program 94.3%

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

      1. associate-*l/ 89.2%

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

      2. +-commutative 89.2%

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

      3. unpow2 89.2%

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

      4. unpow2 89.2%

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

      5. hypot-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in th around 0 90.2%

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

      1. expm1-log1p-u 90.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 20.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 20.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l/ 90.2%

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

      5. associate-*r/ 98.8%

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

   8. Simplified 98.8%

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

   if 5.0000000000000002e-23 < (sin.f64 th) < 0.984999999999999987

   1. Initial program 88.8%

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

      1. +-commutative 88.8%

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

      2. unpow2 88.8%

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

      3. unpow2 88.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 41.3%

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

   if 0.984999999999999987 < (sin.f64 th)

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. unpow2 91.3%

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

      3. unpow2 91.3%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 42.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{\sin th \cdot ky}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 33.6%

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

      2. sqrt-unprod 27.1%

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

      3. pow2 27.1%

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

      4. *-commutative 27.1%

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

   8. Applied egg-rr 27.1%

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

      1. unpow2 27.1%

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

      2. rem-sqrt-square 51.4%

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

      3. *-commutative 51.4%

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

   10. Simplified 51.4%

       \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\left|\sin th \cdot ky\right|}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{\left|\sin th \cdot ky\right|}}\\ \end{array} \]

Alternative 7: 58.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-23}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{\left|\sin th \cdot ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (fabs (* (sin ky) (/ (sin th) (sin kx))))
   (if (<= (sin th) 5e-23)
     (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
     (if (<= (sin th) 0.985)
       (sin th)
       (/ 1.0 (/ (sin kx) (fabs (* (sin th) ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.05) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(th) <= 5e-23) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else if (sin(th) <= 0.985) {
		tmp = sin(th);
	} else {
		tmp = 1.0 / (sin(kx) / fabs((sin(th) * ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.05) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(th) <= 5e-23) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(th) <= 0.985) {
		tmp = Math.sin(th);
	} else {
		tmp = 1.0 / (Math.sin(kx) / Math.abs((Math.sin(th) * ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.05:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(th) <= 5e-23:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(th) <= 0.985:
		tmp = math.sin(th)
	else:
		tmp = 1.0 / (math.sin(kx) / math.fabs((math.sin(th) * ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.05)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(th) <= 5e-23)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	elseif (sin(th) <= 0.985)
		tmp = sin(th);
	else
		tmp = Float64(1.0 / Float64(sin(kx) / abs(Float64(sin(th) * ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.05)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(th) <= 5e-23)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	elseif (sin(th) <= 0.985)
		tmp = sin(th);
	else
		tmp = 1.0 / (sin(kx) / abs((sin(th) * ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 5e-23], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.985], N[Sin[th], $MachinePrecision], N[(1.0 / N[(N[Sin[kx], $MachinePrecision] / N[Abs[N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.050000000000000003

   1. Initial program 90.1%

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

      1. +-commutative 90.1%

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

      2. unpow2 90.1%

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

      3. unpow2 90.1%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 25.4%

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

      1. add-sqr-sqrt 15.7%

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

      2. sqrt-unprod 24.7%

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

      3. pow2 24.7%

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

      4. *-commutative 24.7%

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

   6. Applied egg-rr 24.7%

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

      1. unpow2 24.7%

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

      2. rem-sqrt-square 29.9%

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

      3. associate-*r/ 30.0%

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

      4. associate-/l* 30.0%

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

      5. associate-/r/ 30.0%

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

   8. Simplified 30.0%

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

   if -0.050000000000000003 < (sin.f64 th) < 5.0000000000000002e-23

   1. Initial program 94.3%

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

      1. associate-*l/ 89.2%

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

      2. +-commutative 89.2%

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

      3. unpow2 89.2%

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

      4. unpow2 89.2%

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

      5. hypot-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in th around 0 90.2%

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

      1. associate-/l* 98.8%

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

      2. associate-/r/ 98.8%

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

   6. Applied egg-rr 98.8%

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

   if 5.0000000000000002e-23 < (sin.f64 th) < 0.984999999999999987

   1. Initial program 88.8%

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

      1. +-commutative 88.8%

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

      2. unpow2 88.8%

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

      3. unpow2 88.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 41.3%

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

   if 0.984999999999999987 < (sin.f64 th)

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. unpow2 91.3%

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

      3. unpow2 91.3%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 42.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{\sin th \cdot ky}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 33.6%

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

      2. sqrt-unprod 27.1%

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

      3. pow2 27.1%

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

      4. *-commutative 27.1%

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

   8. Applied egg-rr 27.1%

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

      1. unpow2 27.1%

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

      2. rem-sqrt-square 51.4%

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

      3. *-commutative 51.4%

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

   10. Simplified 51.4%

       \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\left|\sin th \cdot ky\right|}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-23}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{\left|\sin th \cdot ky\right|}}\\ \end{array} \]

Alternative 8: 58.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{\left|\sin th \cdot ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (fabs (* (sin ky) (/ (sin th) (sin kx))))
   (if (<= (sin th) 5e-23)
     (/ th (/ (hypot (sin ky) (sin kx)) (sin ky)))
     (if (<= (sin th) 0.985)
       (sin th)
       (/ 1.0 (/ (sin kx) (fabs (* (sin th) ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.05) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(th) <= 5e-23) {
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	} else if (sin(th) <= 0.985) {
		tmp = sin(th);
	} else {
		tmp = 1.0 / (sin(kx) / fabs((sin(th) * ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.05) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(th) <= 5e-23) {
		tmp = th / (Math.hypot(Math.sin(ky), Math.sin(kx)) / Math.sin(ky));
	} else if (Math.sin(th) <= 0.985) {
		tmp = Math.sin(th);
	} else {
		tmp = 1.0 / (Math.sin(kx) / Math.abs((Math.sin(th) * ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.05:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(th) <= 5e-23:
		tmp = th / (math.hypot(math.sin(ky), math.sin(kx)) / math.sin(ky))
	elif math.sin(th) <= 0.985:
		tmp = math.sin(th)
	else:
		tmp = 1.0 / (math.sin(kx) / math.fabs((math.sin(th) * ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.05)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(th) <= 5e-23)
		tmp = Float64(th / Float64(hypot(sin(ky), sin(kx)) / sin(ky)));
	elseif (sin(th) <= 0.985)
		tmp = sin(th);
	else
		tmp = Float64(1.0 / Float64(sin(kx) / abs(Float64(sin(th) * ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.05)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(th) <= 5e-23)
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	elseif (sin(th) <= 0.985)
		tmp = sin(th);
	else
		tmp = 1.0 / (sin(kx) / abs((sin(th) * ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 5e-23], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.985], N[Sin[th], $MachinePrecision], N[(1.0 / N[(N[Sin[kx], $MachinePrecision] / N[Abs[N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.050000000000000003

   1. Initial program 90.1%

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

      1. +-commutative 90.1%

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

      2. unpow2 90.1%

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

      3. unpow2 90.1%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 25.4%

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

      1. add-sqr-sqrt 15.7%

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

      2. sqrt-unprod 24.7%

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

      3. pow2 24.7%

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

      4. *-commutative 24.7%

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

   6. Applied egg-rr 24.7%

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

      1. unpow2 24.7%

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

      2. rem-sqrt-square 29.9%

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

      3. associate-*r/ 30.0%

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

      4. associate-/l* 30.0%

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

      5. associate-/r/ 30.0%

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

   8. Simplified 30.0%

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

   if -0.050000000000000003 < (sin.f64 th) < 5.0000000000000002e-23

   1. Initial program 94.3%

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

      1. associate-*l/ 89.2%

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

      2. +-commutative 89.2%

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

      3. unpow2 89.2%

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

      4. unpow2 89.2%

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

      5. hypot-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in th around 0 90.2%

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

      1. expm1-log1p-u 90.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 20.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 20.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l/ 90.2%

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

      5. associate-*r/ 98.8%

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

   8. Simplified 98.8%

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

      1. associate-*r/ 90.2%

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

      2. *-commutative 90.2%

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

      3. associate-/l* 98.8%

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

   10. Applied egg-rr 98.8%

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

   if 5.0000000000000002e-23 < (sin.f64 th) < 0.984999999999999987

   1. Initial program 88.8%

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

      1. +-commutative 88.8%

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

      2. unpow2 88.8%

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

      3. unpow2 88.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 41.3%

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

   if 0.984999999999999987 < (sin.f64 th)

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. unpow2 91.3%

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

      3. unpow2 91.3%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 42.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{\sin th \cdot ky}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 33.6%

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

      2. sqrt-unprod 27.1%

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

      3. pow2 27.1%

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

      4. *-commutative 27.1%

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

   8. Applied egg-rr 27.1%

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

      1. unpow2 27.1%

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

      2. rem-sqrt-square 51.4%

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

      3. *-commutative 51.4%

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

   10. Simplified 51.4%

       \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\left|\sin th \cdot ky\right|}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{\left|\sin th \cdot ky\right|}}\\ \end{array} \]

Alternative 9: 76.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{th}{\frac{t_1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.01)
     (/ th (/ t_1 (sin ky)))
     (if (<= (sin ky) 0.002) (/ (sin th) (* t_1 (/ 1.0 ky))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = th / (t_1 / sin(ky));
	} else if (sin(ky) <= 0.002) {
		tmp = sin(th) / (t_1 * (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	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 (Math.sin(ky) <= -0.01) {
		tmp = th / (t_1 / Math.sin(ky));
	} else if (Math.sin(ky) <= 0.002) {
		tmp = Math.sin(th) / (t_1 * (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = th / (t_1 / math.sin(ky))
	elif math.sin(ky) <= 0.002:
		tmp = math.sin(th) / (t_1 * (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(th / Float64(t_1 / sin(ky)));
	elseif (sin(ky) <= 0.002)
		tmp = Float64(sin(th) / Float64(t_1 * Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = th / (t_1 / sin(ky));
	elseif (sin(ky) <= 0.002)
		tmp = sin(th) / (t_1 * (1.0 / ky));
	else
		tmp = sin(th);
	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[N[Sin[ky], $MachinePrecision], -0.01], N[(th / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.002], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 57.1%

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

      1. expm1-log1p-u 56.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 4.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 57.2%

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

      3. *-commutative 57.2%

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

      4. associate-*l/ 57.1%

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

      5. associate-*r/ 57.1%

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

   8. Simplified 57.1%

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

      1. associate-*r/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-/l* 57.2%

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

   10. Applied egg-rr 57.2%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 2e-3

   1. Initial program 84.1%

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

      1. +-commutative 84.1%

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

      2. unpow2 84.1%

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

      3. unpow2 84.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in ky around 0 98.6%

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

   if 2e-3 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 64.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 72.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{th}{\frac{t_1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.01)
     (/ th (/ t_1 (sin ky)))
     (if (<= (sin ky) 0.002) (/ (* (sin th) ky) t_1) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = th / (t_1 / sin(ky));
	} else if (sin(ky) <= 0.002) {
		tmp = (sin(th) * ky) / t_1;
	} else {
		tmp = sin(th);
	}
	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 (Math.sin(ky) <= -0.01) {
		tmp = th / (t_1 / Math.sin(ky));
	} else if (Math.sin(ky) <= 0.002) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = th / (t_1 / math.sin(ky))
	elif math.sin(ky) <= 0.002:
		tmp = (math.sin(th) * ky) / t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(th / Float64(t_1 / sin(ky)));
	elseif (sin(ky) <= 0.002)
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = th / (t_1 / sin(ky));
	elseif (sin(ky) <= 0.002)
		tmp = (sin(th) * ky) / t_1;
	else
		tmp = sin(th);
	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[N[Sin[ky], $MachinePrecision], -0.01], N[(th / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.002], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 57.1%

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

      1. expm1-log1p-u 56.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 4.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 4.4%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 57.2%

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

      3. *-commutative 57.2%

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

      4. associate-*l/ 57.1%

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

      5. associate-*r/ 57.1%

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

   8. Simplified 57.1%

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

      1. associate-*r/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-/l* 57.2%

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

   10. Applied egg-rr 57.2%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 2e-3

   1. Initial program 84.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. +-commutative 78.9%

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

      3. unpow2 78.9%

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

      4. unpow2 78.9%

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

      5. hypot-def 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in ky around 0 90.0%

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

   if 2e-3 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 64.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 47.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (fabs (sin th))
   (if (<= (sin ky) -4e-220)
     (fabs (* (sin th) (/ ky (sin kx))))
     (if (<= (sin ky) 2e-115) (/ (sin th) (/ (sin kx) ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -4e-220) {
		tmp = fabs((sin(th) * (ky / sin(kx))));
	} else if (sin(ky) <= 2e-115) {
		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) <= (-0.01d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-4d-220)) then
        tmp = abs((sin(th) * (ky / sin(kx))))
    else if (sin(ky) <= 2d-115) 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) <= -0.01) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -4e-220) {
		tmp = Math.abs((Math.sin(th) * (ky / Math.sin(kx))));
	} else if (Math.sin(ky) <= 2e-115) {
		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) <= -0.01:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -4e-220:
		tmp = math.fabs((math.sin(th) * (ky / math.sin(kx))))
	elif math.sin(ky) <= 2e-115:
		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) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -4e-220)
		tmp = abs(Float64(sin(th) * Float64(ky / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		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) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -4e-220)
		tmp = abs((sin(th) * (ky / sin(kx))));
	elseif (sin(ky) <= 2e-115)
		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], -0.01], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -4e-220], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-115], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.1%

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

      3. sqrt-unprod 25.8%

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

      4. pow2 25.8%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 25.8%

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

      2. rem-sqrt-square 33.9%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 33.9%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.0100000000000000002 < (sin.f64 ky) < -3.99999999999999997e-220

   1. Initial program 92.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 95.7%

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

      2. clear-num 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in ky around 0 39.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{\sin th \cdot ky}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. sqrt-unprod 45.1%

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

      3. pow2 45.1%

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

      4. clear-num 45.0%

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

      5. *-un-lft-identity 45.0%

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

      6. times-frac 45.0%

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

      7. /-rgt-identity 45.0%

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

   8. Applied egg-rr 45.0%

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

      1. unpow2 45.0%

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

      2. rem-sqrt-square 56.1%

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

   10. Simplified 56.1%

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

   if -3.99999999999999997e-220 < (sin.f64 ky) < 2.0000000000000001e-115

   1. Initial program 67.3%

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

      1. +-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. unpow2 67.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 58.3%

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

      1. associate-/l* 64.1%

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

   6. Simplified 64.1%

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

   if 2.0000000000000001e-115 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 62.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 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 92.0%

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

   1. associate-*l/ 89.4%

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

   2. associate-*r/ 91.9%

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

   3. +-commutative 91.9%

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

   4. unpow2 91.9%

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

   5. unpow2 91.9%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 13: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.0%

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

   1. +-commutative 92.0%

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

   2. unpow2 92.0%

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

   3. unpow2 92.0%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 14: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-31)
   (fabs (sin th))
   (if (<= (sin ky) 2e-115) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-31) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-115) {
		tmp = sin(th) * (ky / 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) <= (-1d-31)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-115) then
        tmp = sin(th) * (ky / 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) <= -1e-31) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-115) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-31:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-115:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-115)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-115)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-31], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-115], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-31

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.2%

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

      3. sqrt-unprod 25.7%

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

      4. pow2 25.7%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 25.7%

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

      2. rem-sqrt-square 33.2%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 33.2%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -1e-31 < (sin.f64 ky) < 2.0000000000000001e-115

   1. Initial program 78.3%

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

      1. +-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. unpow2 78.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 55.1%

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

   if 2.0000000000000001e-115 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 62.1%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-31)
   (fabs (sin th))
   (if (<= (sin ky) 2e-115) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-31) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-115) {
		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) <= (-1d-31)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-115) 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) <= -1e-31) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-115) {
		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) <= -1e-31:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-115:
		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) <= -1e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-115)
		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) <= -1e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-115)
		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], -1e-31], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-115], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-31

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.2%

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

      3. sqrt-unprod 25.7%

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

      4. pow2 25.7%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 25.7%

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

      2. rem-sqrt-square 33.2%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 33.2%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -1e-31 < (sin.f64 ky) < 2.0000000000000001e-115

   1. Initial program 78.3%

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

      1. +-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. unpow2 78.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 51.8%

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

      1. associate-/l* 55.2%

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

   6. Simplified 55.2%

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

   if 2.0000000000000001e-115 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 62.1%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 34.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{elif}\;ky \leq 2.15 \cdot 10^{+112}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -1e-121)
     t_1
     (if (<= ky 5e-134)
       (/ (sin th) (/ kx ky))
       (if (<= ky 2.15e+112) (sin th) t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -1e-121) {
		tmp = t_1;
	} else if (ky <= 5e-134) {
		tmp = sin(th) / (kx / ky);
	} else if (ky <= 2.15e+112) {
		tmp = sin(th);
	} else {
		tmp = 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 = abs(sin(th))
    if (ky <= (-1d-121)) then
        tmp = t_1
    else if (ky <= 5d-134) then
        tmp = sin(th) / (kx / ky)
    else if (ky <= 2.15d+112) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -1e-121) {
		tmp = t_1;
	} else if (ky <= 5e-134) {
		tmp = Math.sin(th) / (kx / ky);
	} else if (ky <= 2.15e+112) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -1e-121:
		tmp = t_1
	elif ky <= 5e-134:
		tmp = math.sin(th) / (kx / ky)
	elif ky <= 2.15e+112:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -1e-121)
		tmp = t_1;
	elseif (ky <= 5e-134)
		tmp = Float64(sin(th) / Float64(kx / ky));
	elseif (ky <= 2.15e+112)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -1e-121)
		tmp = t_1;
	elseif (ky <= 5e-134)
		tmp = sin(th) / (kx / ky);
	elseif (ky <= 2.15e+112)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -1e-121], t$95$1, If[LessEqual[ky, 5e-134], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.15e+112], N[Sin[th], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -9.9999999999999998e-122 or 2.14999999999999991e112 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 98.4%

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

      2. clear-num 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in kx around 0 20.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   7. Step-by-step derivation

      1. remove-double-div 20.0%

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

      2. add-sqr-sqrt 12.4%

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

      3. sqrt-unprod 29.3%

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

      4. pow2 29.3%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   8. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 29.3%

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

      2. rem-sqrt-square 35.5%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   10. Simplified 35.5%

       \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -9.9999999999999998e-122 < ky < 5.0000000000000003e-134

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. unpow2 71.6%

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

      3. unpow2 71.6%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 88.9%

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

      2. clear-num 88.9%

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

   5. Applied egg-rr 88.9%

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

   6. Taylor expanded in ky around 0 55.5%

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

   7. Taylor expanded in kx around 0 38.4%

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

      1. associate-/l* 43.0%

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

   9. Simplified 43.0%

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

   if 5.0000000000000003e-134 < ky < 2.14999999999999991e112

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1 \cdot 10^{-121}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{elif}\;ky \leq 2.15 \cdot 10^{+112}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]

Alternative 17: 33.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.25e-32)
   (sin th)
   (if (<= ky 3.5e-145) (/ ky (/ (sin kx) th)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.25e-32) {
		tmp = sin(th);
	} else if (ky <= 3.5e-145) {
		tmp = ky / (sin(kx) / 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 (ky <= (-1.25d-32)) then
        tmp = sin(th)
    else if (ky <= 3.5d-145) then
        tmp = ky / (sin(kx) / 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 (ky <= -1.25e-32) {
		tmp = Math.sin(th);
	} else if (ky <= 3.5e-145) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.25e-32:
		tmp = math.sin(th)
	elif ky <= 3.5e-145:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.25e-32)
		tmp = sin(th);
	elseif (ky <= 3.5e-145)
		tmp = Float64(ky / Float64(sin(kx) / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.25e-32)
		tmp = sin(th);
	elseif (ky <= 3.5e-145)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.25e-32], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.5e-145], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.25e-32 or 3.49999999999999997e-145 < 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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 31.9%

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

   if -1.25e-32 < ky < 3.49999999999999997e-145

   1. Initial program 76.3%

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

      1. associate-*l/ 70.8%

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

      2. +-commutative 70.8%

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

      3. unpow2 70.8%

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

      4. unpow2 70.8%

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

      5. hypot-def 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in th around 0 41.0%

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

   5. Taylor expanded in ky around 0 30.5%

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

      1. associate-/l* 34.1%

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

   7. Simplified 34.1%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18: 33.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.25e-32)
   (sin th)
   (if (<= ky 3.8e-145) (/ th (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.25e-32) {
		tmp = sin(th);
	} else if (ky <= 3.8e-145) {
		tmp = 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 (ky <= (-1.25d-32)) then
        tmp = sin(th)
    else if (ky <= 3.8d-145) then
        tmp = 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 (ky <= -1.25e-32) {
		tmp = Math.sin(th);
	} else if (ky <= 3.8e-145) {
		tmp = th / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.25e-32:
		tmp = math.sin(th)
	elif ky <= 3.8e-145:
		tmp = th / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.25e-32)
		tmp = sin(th);
	elseif (ky <= 3.8e-145)
		tmp = Float64(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 (ky <= -1.25e-32)
		tmp = sin(th);
	elseif (ky <= 3.8e-145)
		tmp = th / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.25e-32], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.8e-145], N[(th / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.25e-32 or 3.8000000000000002e-145 < 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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 31.9%

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

   if -1.25e-32 < ky < 3.8000000000000002e-145

   1. Initial program 76.3%

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

      1. +-commutative 76.3%

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

      2. unpow2 76.3%

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

      3. unpow2 76.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*l/ 88.9%

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

      2. clear-num 88.4%

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

   5. Applied egg-rr 88.4%

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

   6. Taylor expanded in ky around 0 50.3%

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

   7. Taylor expanded in th around 0 30.5%

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

      1. associate-/l* 34.2%

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

   9. Simplified 34.2%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19: 34.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -53000000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -53000000000.0)
   (sin th)
   (if (<= ky 4.2e-134) (/ (sin th) (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -53000000000.0) {
		tmp = sin(th);
	} else if (ky <= 4.2e-134) {
		tmp = sin(th) / (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 (ky <= (-53000000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 4.2d-134) then
        tmp = sin(th) / (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 (ky <= -53000000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 4.2e-134) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -53000000000.0:
		tmp = math.sin(th)
	elif ky <= 4.2e-134:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -53000000000.0)
		tmp = sin(th);
	elseif (ky <= 4.2e-134)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -53000000000.0)
		tmp = sin(th);
	elseif (ky <= 4.2e-134)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -53000000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 4.2e-134], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.3e10 or 4.1999999999999998e-134 < 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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 33.0%

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

   if -5.3e10 < ky < 4.1999999999999998e-134

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. unpow2 78.8%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*l/ 90.0%

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

      2. clear-num 89.6%

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

   5. Applied egg-rr 89.6%

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

   6. Taylor expanded in ky around 0 48.2%

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

   7. Taylor expanded in kx around 0 32.9%

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

      1. associate-/l* 36.2%

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

   9. Simplified 36.2%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -53000000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 20: 25.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq -3.4 \cdot 10^{-111} \lor \neg \left(kx \leq 6.2 \cdot 10^{-47}\right):\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= kx -3.4e-111) (not (<= kx 6.2e-47))) (sqrt (* th th)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((kx <= -3.4e-111) || !(kx <= 6.2e-47)) {
		tmp = sqrt((th * 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 ((kx <= (-3.4d-111)) .or. (.not. (kx <= 6.2d-47))) then
        tmp = sqrt((th * 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 ((kx <= -3.4e-111) || !(kx <= 6.2e-47)) {
		tmp = Math.sqrt((th * th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (kx <= -3.4e-111) or not (kx <= 6.2e-47):
		tmp = math.sqrt((th * th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((kx <= -3.4e-111) || !(kx <= 6.2e-47))
		tmp = sqrt(Float64(th * th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((kx <= -3.4e-111) || ~((kx <= 6.2e-47)))
		tmp = sqrt((th * th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[kx, -3.4e-111], N[Not[LessEqual[kx, 6.2e-47]], $MachinePrecision]], N[Sqrt[N[(th * th), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < -3.39999999999999997e-111 or 6.1999999999999996e-47 < 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. +-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-def 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 98.3%

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

      2. clear-num 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in kx around 0 10.3%

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

   7. Taylor expanded in th around 0 5.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th}}} \]
   8. Step-by-step derivation

      1. remove-double-div 5.6%

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

      2. add-sqr-sqrt 3.5%

         \[\leadsto \color{blue}{\sqrt{th} \cdot \sqrt{th}} \]

      3. sqrt-unprod 18.2%

         \[\leadsto \color{blue}{\sqrt{th \cdot th}} \]

   9. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{\sqrt{th \cdot th}} \]

   if -3.39999999999999997e-111 < kx < 6.1999999999999996e-47

   1. Initial program 82.2%

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

      1. +-commutative 82.2%

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

      2. unpow2 82.2%

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

      3. unpow2 82.2%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 44.4%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq -3.4 \cdot 10^{-111} \lor \neg \left(kx \leq 6.2 \cdot 10^{-47}\right):\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 21: 24.8% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return 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(th)
end function

Java

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

Python

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

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 92.0%

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

   1. +-commutative 92.0%

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

   2. unpow2 92.0%

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

   3. unpow2 92.0%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in kx around 0 25.1%

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

5. Final simplification 25.1%

   \[\leadsto \sin th \]

Alternative 22: 14.7% accurate, 78.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ 1.0 (+ (/ 1.0 th) (* th 0.16666666666666666))))

C

double code(double kx, double ky, double th) {
	return 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 1.0d0 / ((1.0d0 / th) + (th * 0.16666666666666666d0))
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
}

Python

def code(kx, ky, th):
	return 1.0 / ((1.0 / th) + (th * 0.16666666666666666))

Julia

function code(kx, ky, th)
	return Float64(1.0 / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

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

   1. +-commutative 92.0%

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

   2. unpow2 92.0%

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

   3. unpow2 92.0%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 95.4%

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

   2. clear-num 95.1%

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

5. Applied egg-rr 95.1%

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

6. Taylor expanded in kx around 0 25.1%

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

7. Taylor expanded in th around 0 13.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
8. Step-by-step derivation

   1. *-commutative 13.6%

      \[\leadsto \frac{1}{\frac{1}{th} + \color{blue}{th \cdot 0.16666666666666666}} \]

9. Simplified 13.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{th} + th \cdot 0.16666666666666666}} \]

10. Final simplification 13.6%

    \[\leadsto \frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666} \]

Alternative 23: 14.1% 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 92.0%

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

   1. associate-*l/ 89.4%

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

   2. +-commutative 89.4%

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

   3. unpow2 89.4%

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

   4. unpow2 89.4%

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

   5. hypot-def 95.4%

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

3. Simplified 95.4%

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

4. Taylor expanded in th around 0 46.7%

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

5. Taylor expanded in kx around 0 13.1%

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

6. Final simplification 13.1%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023194 
(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)))