Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 21.0s

Alternatives: 17

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: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. +-commutative 89.8%

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

   2. unpow2 89.8%

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

   3. unpow2 89.8%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin th \leq -0.76:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.64:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin th \leq -0.001:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 10^{-14}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin th) -0.76)
     t_1
     (if (<= (sin th) -0.64)
       (/ (sin th) (+ 1.0 (/ (* 0.5 (* kx kx)) (pow (sin ky) 2.0))))
       (if (<= (sin th) -0.001)
         t_1
         (if (<= (sin th) 1e-14)
           (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
           (/ (fabs (* (sin ky) (sin th))) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(th) <= -0.76) {
		tmp = t_1;
	} else if (sin(th) <= -0.64) {
		tmp = sin(th) / (1.0 + ((0.5 * (kx * kx)) / pow(sin(ky), 2.0)));
	} else if (sin(th) <= -0.001) {
		tmp = t_1;
	} else if (sin(th) <= 1e-14) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = fabs((sin(ky) * sin(th))) / sin(ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (Math.sin(th) <= -0.76) {
		tmp = t_1;
	} else if (Math.sin(th) <= -0.64) {
		tmp = Math.sin(th) / (1.0 + ((0.5 * (kx * kx)) / Math.pow(Math.sin(ky), 2.0)));
	} else if (Math.sin(th) <= -0.001) {
		tmp = t_1;
	} else if (Math.sin(th) <= 1e-14) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.abs((Math.sin(ky) * Math.sin(th))) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if math.sin(th) <= -0.76:
		tmp = t_1
	elif math.sin(th) <= -0.64:
		tmp = math.sin(th) / (1.0 + ((0.5 * (kx * kx)) / math.pow(math.sin(ky), 2.0)))
	elif math.sin(th) <= -0.001:
		tmp = t_1
	elif math.sin(th) <= 1e-14:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.fabs((math.sin(ky) * math.sin(th))) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (sin(th) <= -0.76)
		tmp = t_1;
	elseif (sin(th) <= -0.64)
		tmp = Float64(sin(th) / Float64(1.0 + Float64(Float64(0.5 * Float64(kx * kx)) / (sin(ky) ^ 2.0))));
	elseif (sin(th) <= -0.001)
		tmp = t_1;
	elseif (sin(th) <= 1e-14)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(abs(Float64(sin(ky) * sin(th))) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (sin(th) <= -0.76)
		tmp = t_1;
	elseif (sin(th) <= -0.64)
		tmp = sin(th) / (1.0 + ((0.5 * (kx * kx)) / (sin(ky) ^ 2.0)));
	elseif (sin(th) <= -0.001)
		tmp = t_1;
	elseif (sin(th) <= 1e-14)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = abs((sin(ky) * sin(th))) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.76], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], -0.64], 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[th], $MachinePrecision], -0.001], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], 1e-14], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.76000000000000001 or -0.640000000000000013 < (sin.f64 th) < -1e-3

   1. Initial program 84.0%

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

      1. associate-*l/ 84.0%

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

      2. associate-*r/ 83.8%

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

      3. +-commutative 83.8%

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

      4. unpow2 83.8%

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

      5. unpow2 83.8%

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

      6. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in kx around 0 14.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 35.5%

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

      3. pow2 35.5%

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

   6. Applied egg-rr 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. *-inverses 35.6%

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

      4. /-rgt-identity 35.6%

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

      5. unpow2 35.6%

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

      6. rem-sqrt-square 35.6%

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

   8. Simplified 35.6%

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

   if -0.76000000000000001 < (sin.f64 th) < -0.640000000000000013

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/l* 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 84.3%

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

      1. associate-*r/ 84.3%

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

      2. unpow2 84.3%

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

   8. Simplified 84.3%

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

   if -1e-3 < (sin.f64 th) < 9.99999999999999999e-15

   1. Initial program 93.8%

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

      1. associate-*l/ 86.5%

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

      2. +-commutative 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. hypot-def 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in th around 0 89.8%

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

      1. expm1-log1p-u 89.8%

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

      2. expm1-udef 20.0%

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

      3. *-un-lft-identity 20.0%

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

      4. times-frac 20.0%

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

      5. /-rgt-identity 20.0%

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

   6. Applied egg-rr 20.0%

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

      1. expm1-def 99.8%

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

      2. expm1-log1p 99.8%

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

   8. Simplified 99.8%

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

   if 9.99999999999999999e-15 < (sin.f64 th)

   1. Initial program 86.0%

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

      1. associate-*l/ 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.1%

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

      4. unpow2 86.1%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 26.1%

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

      1. add-sqr-sqrt 25.0%

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

      2. sqrt-unprod 45.3%

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

      3. pow2 45.3%

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

      4. *-commutative 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. unpow2 45.3%

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

      2. rem-sqrt-square 47.5%

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

      3. *-commutative 47.5%

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

   8. Simplified 47.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.76:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq -0.64:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin th \leq -0.001:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 10^{-14}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \end{array} \]

Alternative 3: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin th \leq -0.76:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.64:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin th \leq -0.001:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin th) -0.76)
     t_1
     (if (<= (sin th) -0.64)
       (/ (sin th) (+ 1.0 (/ (* 0.5 (* kx kx)) (pow (sin ky) 2.0))))
       (if (<= (sin th) -0.001)
         t_1
         (if (<= (sin th) 1e-14)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (/ (fabs (* (sin ky) (sin th))) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(th) <= -0.76) {
		tmp = t_1;
	} else if (sin(th) <= -0.64) {
		tmp = sin(th) / (1.0 + ((0.5 * (kx * kx)) / pow(sin(ky), 2.0)));
	} else if (sin(th) <= -0.001) {
		tmp = t_1;
	} else if (sin(th) <= 1e-14) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = fabs((sin(ky) * sin(th))) / sin(ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (Math.sin(th) <= -0.76) {
		tmp = t_1;
	} else if (Math.sin(th) <= -0.64) {
		tmp = Math.sin(th) / (1.0 + ((0.5 * (kx * kx)) / Math.pow(Math.sin(ky), 2.0)));
	} else if (Math.sin(th) <= -0.001) {
		tmp = t_1;
	} else if (Math.sin(th) <= 1e-14) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = Math.abs((Math.sin(ky) * Math.sin(th))) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if math.sin(th) <= -0.76:
		tmp = t_1
	elif math.sin(th) <= -0.64:
		tmp = math.sin(th) / (1.0 + ((0.5 * (kx * kx)) / math.pow(math.sin(ky), 2.0)))
	elif math.sin(th) <= -0.001:
		tmp = t_1
	elif math.sin(th) <= 1e-14:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = math.fabs((math.sin(ky) * math.sin(th))) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (sin(th) <= -0.76)
		tmp = t_1;
	elseif (sin(th) <= -0.64)
		tmp = Float64(sin(th) / Float64(1.0 + Float64(Float64(0.5 * Float64(kx * kx)) / (sin(ky) ^ 2.0))));
	elseif (sin(th) <= -0.001)
		tmp = t_1;
	elseif (sin(th) <= 1e-14)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(abs(Float64(sin(ky) * sin(th))) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (sin(th) <= -0.76)
		tmp = t_1;
	elseif (sin(th) <= -0.64)
		tmp = sin(th) / (1.0 + ((0.5 * (kx * kx)) / (sin(ky) ^ 2.0)));
	elseif (sin(th) <= -0.001)
		tmp = t_1;
	elseif (sin(th) <= 1e-14)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = abs((sin(ky) * sin(th))) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.76], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], -0.64], 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[th], $MachinePrecision], -0.001], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], 1e-14], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.76000000000000001 or -0.640000000000000013 < (sin.f64 th) < -1e-3

   1. Initial program 84.0%

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

      1. associate-*l/ 84.0%

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

      2. associate-*r/ 83.8%

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

      3. +-commutative 83.8%

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

      4. unpow2 83.8%

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

      5. unpow2 83.8%

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

      6. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in kx around 0 14.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 35.5%

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

      3. pow2 35.5%

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

   6. Applied egg-rr 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. *-inverses 35.6%

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

      4. /-rgt-identity 35.6%

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

      5. unpow2 35.6%

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

      6. rem-sqrt-square 35.6%

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

   8. Simplified 35.6%

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

   if -0.76000000000000001 < (sin.f64 th) < -0.640000000000000013

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/l* 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 84.3%

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

      1. associate-*r/ 84.3%

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

      2. unpow2 84.3%

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

   8. Simplified 84.3%

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

   if -1e-3 < (sin.f64 th) < 9.99999999999999999e-15

   1. Initial program 93.8%

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

      1. associate-*l/ 86.5%

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

      2. +-commutative 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. hypot-def 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in th around 0 89.8%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 9.99999999999999999e-15 < (sin.f64 th)

   1. Initial program 86.0%

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

      1. associate-*l/ 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.1%

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

      4. unpow2 86.1%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 26.1%

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

      1. add-sqr-sqrt 25.0%

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

      2. sqrt-unprod 45.3%

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

      3. pow2 45.3%

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

      4. *-commutative 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. unpow2 45.3%

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

      2. rem-sqrt-square 47.5%

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

      3. *-commutative 47.5%

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

   8. Simplified 47.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.76:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq -0.64:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin th \leq -0.001:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \end{array} \]

Alternative 4: 47.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.71:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.97)
   (fabs (sin th))
   (if (<= (sin ky) -0.71)
     (- th)
     (if (<= (sin ky) -0.02)
       (* (sin ky) (fabs (/ (sin th) (sin ky))))
       (if (<= (sin ky) -5e-181)
         (/ (* ky th) (hypot (sin ky) (sin kx)))
         (if (<= (sin ky) 5e-206) (* (sin th) (/ ky (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.97) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -0.71) {
		tmp = -th;
	} else if (sin(ky) <= -0.02) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else if (sin(ky) <= -5e-181) {
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= 5e-206) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.97) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -0.71) {
		tmp = -th;
	} else if (Math.sin(ky) <= -0.02) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else if (Math.sin(ky) <= -5e-181) {
		tmp = (ky * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= 5e-206) {
		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) <= -0.97:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -0.71:
		tmp = -th
	elif math.sin(ky) <= -0.02:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	elif math.sin(ky) <= -5e-181:
		tmp = (ky * th) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= 5e-206:
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.71)
		tmp = Float64(-th);
	elseif (sin(ky) <= -0.02)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	elseif (sin(ky) <= -5e-181)
		tmp = Float64(Float64(ky * th) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 5e-206)
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.71)
		tmp = -th;
	elseif (sin(ky) <= -0.02)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	elseif (sin(ky) <= -5e-181)
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= 5e-206)
		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], -0.97], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.71], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-181], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-206], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (sin.f64 ky) < -0.96999999999999997

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 1.7%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 58.7%

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

      3. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/r/ 58.7%

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

      3. *-inverses 58.7%

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

      4. /-rgt-identity 58.7%

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

      5. unpow2 58.7%

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

      6. rem-sqrt-square 58.7%

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

   8. Simplified 58.7%

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

   if -0.96999999999999997 < (sin.f64 ky) < -0.70999999999999996

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 3.5%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 18.9%

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

      3. pow2 18.9%

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

   6. Applied egg-rr 18.9%

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

      1. *-commutative 18.9%

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

      2. associate-/r/ 18.9%

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

      3. *-inverses 18.9%

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

      4. /-rgt-identity 18.9%

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

   8. Simplified 18.9%

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

   9. Taylor expanded in th around 0 9.0%

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

       1. unpow2 9.0%

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

   11. Simplified 9.0%

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

   12. Taylor expanded in th around -inf 52.1%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 52.1%

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

   14. Simplified 52.1%

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

   if -0.70999999999999996 < (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. unpow2 99.8%

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

      5. unpow2 99.8%

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

      6. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 36.1%

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

      3. pow2 36.1%

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

   6. Applied egg-rr 36.1%

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

      1. unpow2 36.1%

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

      2. rem-sqrt-square 39.7%

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

   8. Simplified 39.7%

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

   if -0.0200000000000000004 < (sin.f64 ky) < -5.0000000000000001e-181

   1. Initial program 97.0%

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

      1. associate-*l/ 88.2%

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

      2. +-commutative 88.2%

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

      3. unpow2 88.2%

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

      4. unpow2 88.2%

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

      5. hypot-def 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in th around 0 46.4%

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

   5. Taylor expanded in ky around 0 46.4%

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

   if -5.0000000000000001e-181 < (sin.f64 ky) < 5e-206

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. hypot-def 99.7%

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

   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 63.2%

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

   if 5e-206 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 94.5%

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

      3. +-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. unpow2 94.5%

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

      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. Taylor expanded in kx around 0 57.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.71:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 47.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.71:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.97)
   (fabs (sin th))
   (if (<= (sin ky) -0.71)
     (- th)
     (if (<= (sin ky) -0.02)
       (/ (fabs (* (sin ky) (sin th))) (sin ky))
       (if (<= (sin ky) -5e-181)
         (/ (* ky th) (hypot (sin ky) (sin kx)))
         (if (<= (sin ky) 5e-206) (* (sin th) (/ ky (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.97) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -0.71) {
		tmp = -th;
	} else if (sin(ky) <= -0.02) {
		tmp = fabs((sin(ky) * sin(th))) / sin(ky);
	} else if (sin(ky) <= -5e-181) {
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= 5e-206) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.97) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -0.71) {
		tmp = -th;
	} else if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs((Math.sin(ky) * Math.sin(th))) / Math.sin(ky);
	} else if (Math.sin(ky) <= -5e-181) {
		tmp = (ky * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= 5e-206) {
		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) <= -0.97:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -0.71:
		tmp = -th
	elif math.sin(ky) <= -0.02:
		tmp = math.fabs((math.sin(ky) * math.sin(th))) / math.sin(ky)
	elif math.sin(ky) <= -5e-181:
		tmp = (ky * th) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= 5e-206:
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.71)
		tmp = Float64(-th);
	elseif (sin(ky) <= -0.02)
		tmp = Float64(abs(Float64(sin(ky) * sin(th))) / sin(ky));
	elseif (sin(ky) <= -5e-181)
		tmp = Float64(Float64(ky * th) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 5e-206)
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.71)
		tmp = -th;
	elseif (sin(ky) <= -0.02)
		tmp = abs((sin(ky) * sin(th))) / sin(ky);
	elseif (sin(ky) <= -5e-181)
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= 5e-206)
		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], -0.97], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.71], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-181], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-206], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (sin.f64 ky) < -0.96999999999999997

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 1.7%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 58.7%

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

      3. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/r/ 58.7%

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

      3. *-inverses 58.7%

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

      4. /-rgt-identity 58.7%

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

      5. unpow2 58.7%

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

      6. rem-sqrt-square 58.7%

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

   8. Simplified 58.7%

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

   if -0.96999999999999997 < (sin.f64 ky) < -0.70999999999999996

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 3.5%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 18.9%

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

      3. pow2 18.9%

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

   6. Applied egg-rr 18.9%

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

      1. *-commutative 18.9%

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

      2. associate-/r/ 18.9%

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

      3. *-inverses 18.9%

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

      4. /-rgt-identity 18.9%

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

   8. Simplified 18.9%

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

   9. Taylor expanded in th around 0 9.0%

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

       1. unpow2 9.0%

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

   11. Simplified 9.0%

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

   12. Taylor expanded in th around -inf 52.1%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 52.1%

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

   14. Simplified 52.1%

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

   if -0.70999999999999996 < (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 36.2%

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

      3. pow2 36.2%

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

      4. *-commutative 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. unpow2 36.2%

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

      2. rem-sqrt-square 39.8%

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

      3. *-commutative 39.8%

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

   8. Simplified 39.8%

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

   if -0.0200000000000000004 < (sin.f64 ky) < -5.0000000000000001e-181

   1. Initial program 97.0%

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

      1. associate-*l/ 88.2%

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

      2. +-commutative 88.2%

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

      3. unpow2 88.2%

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

      4. unpow2 88.2%

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

      5. hypot-def 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in th around 0 46.4%

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

   5. Taylor expanded in ky around 0 46.4%

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

   if -5.0000000000000001e-181 < (sin.f64 ky) < 5e-206

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. hypot-def 99.7%

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

   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 63.2%

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

   if 5e-206 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 94.5%

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

      3. +-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. unpow2 94.5%

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

      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. Taylor expanded in kx around 0 57.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.71:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 43.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.97)
   (fabs (sin th))
   (if (<= (sin ky) -5e-30)
     (- th)
     (if (<= (sin ky) -2e-189)
       (sqrt (pow (sin th) 2.0))
       (if (<= (sin ky) 5e-206)
         (/ (sin ky) (/ (sin kx) (sin th)))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.97) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-30) {
		tmp = -th;
	} else if (sin(ky) <= -2e-189) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 5e-206) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.97d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-5d-30)) then
        tmp = -th
    else if (sin(ky) <= (-2d-189)) then
        tmp = sqrt((sin(th) ** 2.0d0))
    else if (sin(ky) <= 5d-206) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.97) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-30) {
		tmp = -th;
	} else if (Math.sin(ky) <= -2e-189) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 5e-206) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.97:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-30:
		tmp = -th
	elif math.sin(ky) <= -2e-189:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 5e-206:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-30)
		tmp = Float64(-th);
	elseif (sin(ky) <= -2e-189)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 5e-206)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-30)
		tmp = -th;
	elseif (sin(ky) <= -2e-189)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 5e-206)
		tmp = sin(ky) / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.97], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-30], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-189], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-206], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.96999999999999997

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 1.7%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 58.7%

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

      3. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/r/ 58.7%

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

      3. *-inverses 58.7%

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

      4. /-rgt-identity 58.7%

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

      5. unpow2 58.7%

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

      6. rem-sqrt-square 58.7%

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

   8. Simplified 58.7%

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

   if -0.96999999999999997 < (sin.f64 ky) < -4.99999999999999972e-30

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 21.8%

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

      3. pow2 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. *-commutative 21.8%

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

      2. associate-/r/ 21.8%

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

      3. *-inverses 21.8%

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

      4. /-rgt-identity 21.8%

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

   8. Simplified 21.8%

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

   9. Taylor expanded in th around 0 11.4%

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

       1. unpow2 11.4%

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

   11. Simplified 11.4%

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

   12. Taylor expanded in th around -inf 39.5%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 39.5%

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

   14. Simplified 39.5%

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

   if -4.99999999999999972e-30 < (sin.f64 ky) < -2.00000000000000014e-189

   1. Initial program 91.3%

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

      1. associate-*l/ 81.9%

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

      2. associate-*r/ 91.0%

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

      3. +-commutative 91.0%

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

      4. unpow2 91.0%

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

      5. unpow2 91.0%

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

      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. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 47.0%

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

      3. pow2 47.0%

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

   6. Applied egg-rr 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-/r/ 47.1%

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

      3. *-inverses 47.1%

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

      4. /-rgt-identity 47.1%

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

   8. Simplified 47.1%

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

   if -2.00000000000000014e-189 < (sin.f64 ky) < 5e-206

   1. Initial program 66.2%

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

      1. associate-/r/ 66.2%

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

      2. +-commutative 66.2%

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

      3. unpow2 66.2%

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

      4. unpow2 66.2%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 64.2%

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

   if 5e-206 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 94.5%

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

      3. +-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. unpow2 94.5%

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

      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. Taylor expanded in kx around 0 57.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t_1}\\ \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{elif}\;\sin th \leq 0.75:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))) (t_2 (/ (* ky (sin th)) t_1)))
   (if (<= (sin th) -0.02)
     t_2
     (if (<= (sin th) 2e-22)
       (* (/ (sin ky) t_1) th)
       (if (<= (sin th) 0.75)
         t_2
         (/ (fabs (* (sin ky) (sin th))) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (ky * sin(th)) / t_1;
	double tmp;
	if (sin(th) <= -0.02) {
		tmp = t_2;
	} else if (sin(th) <= 2e-22) {
		tmp = (sin(ky) / t_1) * th;
	} else if (sin(th) <= 0.75) {
		tmp = t_2;
	} else {
		tmp = fabs((sin(ky) * sin(th))) / sin(ky);
	}
	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 t_2 = (ky * Math.sin(th)) / t_1;
	double tmp;
	if (Math.sin(th) <= -0.02) {
		tmp = t_2;
	} else if (Math.sin(th) <= 2e-22) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (Math.sin(th) <= 0.75) {
		tmp = t_2;
	} else {
		tmp = Math.abs((Math.sin(ky) * Math.sin(th))) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = (ky * math.sin(th)) / t_1
	tmp = 0
	if math.sin(th) <= -0.02:
		tmp = t_2
	elif math.sin(th) <= 2e-22:
		tmp = (math.sin(ky) / t_1) * th
	elif math.sin(th) <= 0.75:
		tmp = t_2
	else:
		tmp = math.fabs((math.sin(ky) * math.sin(th))) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(ky * sin(th)) / t_1)
	tmp = 0.0
	if (sin(th) <= -0.02)
		tmp = t_2;
	elseif (sin(th) <= 2e-22)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (sin(th) <= 0.75)
		tmp = t_2;
	else
		tmp = Float64(abs(Float64(sin(ky) * sin(th))) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = (ky * sin(th)) / t_1;
	tmp = 0.0;
	if (sin(th) <= -0.02)
		tmp = t_2;
	elseif (sin(th) <= 2e-22)
		tmp = (sin(ky) / t_1) * th;
	elseif (sin(th) <= 0.75)
		tmp = t_2;
	else
		tmp = abs((sin(ky) * sin(th))) / sin(ky);
	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]}, Block[{t$95$2 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], t$95$2, If[LessEqual[N[Sin[th], $MachinePrecision], 2e-22], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.75], t$95$2, N[(N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -0.0200000000000000004 or 2.0000000000000001e-22 < (sin.f64 th) < 0.75

   1. Initial program 81.9%

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

      1. associate-*l/ 81.8%

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

      2. +-commutative 81.8%

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

      3. unpow2 81.8%

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

      4. unpow2 81.8%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 59.8%

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

   if -0.0200000000000000004 < (sin.f64 th) < 2.0000000000000001e-22

   1. Initial program 94.5%

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

      1. associate-*l/ 87.3%

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

      2. +-commutative 87.3%

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

      3. unpow2 87.3%

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

      4. unpow2 87.3%

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

      5. hypot-def 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in th around 0 88.4%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 98.3%

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

   6. Applied egg-rr 98.3%

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

   if 0.75 < (sin.f64 th)

   1. Initial program 95.9%

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

      1. associate-*l/ 96.2%

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

      2. +-commutative 96.2%

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

      3. unpow2 96.2%

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

      4. unpow2 96.2%

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

      5. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 32.2%

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

      1. add-sqr-sqrt 31.2%

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

      2. sqrt-unprod 52.3%

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

      3. pow2 52.3%

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

      4. *-commutative 52.3%

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

   6. Applied egg-rr 52.3%

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

      1. unpow2 52.3%

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

      2. rem-sqrt-square 55.7%

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

      3. *-commutative 55.7%

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

   8. Simplified 55.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin th \leq 0.75:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin ky \cdot \sin th\right|}{\sin ky}\\ \end{array} \]

Alternative 8: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -0.04:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin ky) -0.97)
     t_1
     (if (<= (sin ky) -0.04)
       (- th)
       (if (<= (sin ky) -5e-127)
         t_1
         (if (<= (sin ky) 5e-206) (* (sin th) (/ ky (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(ky) <= -0.97) {
		tmp = t_1;
	} else if (sin(ky) <= -0.04) {
		tmp = -th;
	} else if (sin(ky) <= -5e-127) {
		tmp = t_1;
	} else if (sin(ky) <= 5e-206) {
		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) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (sin(ky) <= (-0.97d0)) then
        tmp = t_1
    else if (sin(ky) <= (-0.04d0)) then
        tmp = -th
    else if (sin(ky) <= (-5d-127)) then
        tmp = t_1
    else if (sin(ky) <= 5d-206) 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 t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (Math.sin(ky) <= -0.97) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -0.04) {
		tmp = -th;
	} else if (Math.sin(ky) <= -5e-127) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 5e-206) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if math.sin(ky) <= -0.97:
		tmp = t_1
	elif math.sin(ky) <= -0.04:
		tmp = -th
	elif math.sin(ky) <= -5e-127:
		tmp = t_1
	elif math.sin(ky) <= 5e-206:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (sin(ky) <= -0.97)
		tmp = t_1;
	elseif (sin(ky) <= -0.04)
		tmp = Float64(-th);
	elseif (sin(ky) <= -5e-127)
		tmp = t_1;
	elseif (sin(ky) <= 5e-206)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (sin(ky) <= -0.97)
		tmp = t_1;
	elseif (sin(ky) <= -0.04)
		tmp = -th;
	elseif (sin(ky) <= -5e-127)
		tmp = t_1;
	elseif (sin(ky) <= 5e-206)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.97], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.04], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-127], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-206], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.96999999999999997 or -0.0400000000000000008 < (sin.f64 ky) < -4.9999999999999997e-127

   1. Initial program 99.7%

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

      1. associate-*l/ 96.0%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 41.3%

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

      3. pow2 41.3%

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

   6. Applied egg-rr 41.3%

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

      1. *-commutative 41.3%

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

      2. associate-/r/ 41.4%

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

      3. *-inverses 41.4%

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

      4. /-rgt-identity 41.4%

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

      5. unpow2 41.4%

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

      6. rem-sqrt-square 43.8%

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

   8. Simplified 43.8%

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

   if -0.96999999999999997 < (sin.f64 ky) < -0.0400000000000000008

   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/ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

   6. Applied egg-rr 22.3%

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

      1. *-commutative 22.3%

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

      2. associate-/r/ 22.3%

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

      3. *-inverses 22.3%

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

      4. /-rgt-identity 22.3%

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

   8. Simplified 22.3%

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

   9. Taylor expanded in th around 0 12.2%

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

       1. unpow2 12.2%

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

   11. Simplified 12.2%

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

   12. Taylor expanded in th around -inf 42.5%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 42.5%

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

   14. Simplified 42.5%

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

   if -4.9999999999999997e-127 < (sin.f64 ky) < 5e-206

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

      2. unpow2 68.9%

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

      3. unpow2 68.9%

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

      4. hypot-def 99.8%

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

   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 ky around 0 60.1%

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

   if 5e-206 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 94.5%

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

      3. +-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. unpow2 94.5%

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

      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. Taylor expanded in kx around 0 57.3%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.04:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.97)
   (fabs (sin th))
   (if (<= (sin ky) -5e-30)
     (- th)
     (if (<= (sin ky) -2e-189)
       (sqrt (pow (sin th) 2.0))
       (if (<= (sin ky) 5e-206) (* (sin th) (/ ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.97) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-30) {
		tmp = -th;
	} else if (sin(ky) <= -2e-189) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 5e-206) {
		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) <= (-0.97d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-5d-30)) then
        tmp = -th
    else if (sin(ky) <= (-2d-189)) then
        tmp = sqrt((sin(th) ** 2.0d0))
    else if (sin(ky) <= 5d-206) 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) <= -0.97) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-30) {
		tmp = -th;
	} else if (Math.sin(ky) <= -2e-189) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 5e-206) {
		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) <= -0.97:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-30:
		tmp = -th
	elif math.sin(ky) <= -2e-189:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 5e-206:
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-30)
		tmp = Float64(-th);
	elseif (sin(ky) <= -2e-189)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 5e-206)
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-30)
		tmp = -th;
	elseif (sin(ky) <= -2e-189)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 5e-206)
		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], -0.97], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-30], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-189], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-206], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.96999999999999997

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 1.7%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 58.7%

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

      3. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/r/ 58.7%

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

      3. *-inverses 58.7%

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

      4. /-rgt-identity 58.7%

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

      5. unpow2 58.7%

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

      6. rem-sqrt-square 58.7%

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

   8. Simplified 58.7%

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

   if -0.96999999999999997 < (sin.f64 ky) < -4.99999999999999972e-30

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 21.8%

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

      3. pow2 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. *-commutative 21.8%

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

      2. associate-/r/ 21.8%

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

      3. *-inverses 21.8%

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

      4. /-rgt-identity 21.8%

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

   8. Simplified 21.8%

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

   9. Taylor expanded in th around 0 11.4%

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

       1. unpow2 11.4%

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

   11. Simplified 11.4%

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

   12. Taylor expanded in th around -inf 39.5%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 39.5%

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

   14. Simplified 39.5%

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

   if -4.99999999999999972e-30 < (sin.f64 ky) < -2.00000000000000014e-189

   1. Initial program 91.3%

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

      1. associate-*l/ 81.9%

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

      2. associate-*r/ 91.0%

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

      3. +-commutative 91.0%

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

      4. unpow2 91.0%

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

      5. unpow2 91.0%

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

      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. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 47.0%

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

      3. pow2 47.0%

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

   6. Applied egg-rr 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-/r/ 47.1%

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

      3. *-inverses 47.1%

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

      4. /-rgt-identity 47.1%

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

   8. Simplified 47.1%

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

   if -2.00000000000000014e-189 < (sin.f64 ky) < 5e-206

   1. Initial program 66.2%

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

      1. +-commutative 66.2%

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

      2. unpow2 66.2%

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

      3. unpow2 66.2%

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

      4. hypot-def 99.7%

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

   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 64.1%

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

   if 5e-206 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 94.5%

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

      3. +-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. unpow2 94.5%

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

      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. Taylor expanded in kx around 0 57.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 46.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.04:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.97)
   (fabs (sin th))
   (if (<= (sin ky) -0.04)
     (- th)
     (if (<= (sin ky) -5e-181)
       (/ (* ky th) (hypot (sin ky) (sin kx)))
       (if (<= (sin ky) 5e-206) (* (sin th) (/ ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.97) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -0.04) {
		tmp = -th;
	} else if (sin(ky) <= -5e-181) {
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= 5e-206) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.97) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -0.04) {
		tmp = -th;
	} else if (Math.sin(ky) <= -5e-181) {
		tmp = (ky * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= 5e-206) {
		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) <= -0.97:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -0.04:
		tmp = -th
	elif math.sin(ky) <= -5e-181:
		tmp = (ky * th) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= 5e-206:
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.04)
		tmp = Float64(-th);
	elseif (sin(ky) <= -5e-181)
		tmp = Float64(Float64(ky * th) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 5e-206)
		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) <= -0.97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.04)
		tmp = -th;
	elseif (sin(ky) <= -5e-181)
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= 5e-206)
		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], -0.97], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.04], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-181], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-206], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.96999999999999997

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 1.7%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 58.7%

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

      3. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/r/ 58.7%

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

      3. *-inverses 58.7%

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

      4. /-rgt-identity 58.7%

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

      5. unpow2 58.7%

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

      6. rem-sqrt-square 58.7%

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

   8. Simplified 58.7%

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

   if -0.96999999999999997 < (sin.f64 ky) < -0.0400000000000000008

   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/ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

   6. Applied egg-rr 22.3%

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

      1. *-commutative 22.3%

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

      2. associate-/r/ 22.3%

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

      3. *-inverses 22.3%

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

      4. /-rgt-identity 22.3%

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

   8. Simplified 22.3%

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

   9. Taylor expanded in th around 0 12.2%

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

       1. unpow2 12.2%

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

   11. Simplified 12.2%

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

   12. Taylor expanded in th around -inf 42.5%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 42.5%

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

   14. Simplified 42.5%

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

   if -0.0400000000000000008 < (sin.f64 ky) < -5.0000000000000001e-181

   1. Initial program 97.2%

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

      1. associate-*l/ 88.8%

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

      2. +-commutative 88.8%

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

      3. unpow2 88.8%

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

      4. unpow2 88.8%

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

      5. hypot-def 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in th around 0 44.0%

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

   5. Taylor expanded in ky around 0 44.2%

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

   if -5.0000000000000001e-181 < (sin.f64 ky) < 5e-206

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. hypot-def 99.7%

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

   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 63.2%

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

   if 5e-206 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 94.5%

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

      3. +-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. unpow2 94.5%

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

      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. Taylor expanded in kx around 0 57.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.97:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.04:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 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 89.8%

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

   1. associate-*l/ 86.1%

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

   2. associate-*r/ 89.7%

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

   3. +-commutative 89.7%

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

   4. unpow2 89.7%

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

   5. unpow2 89.7%

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

   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 12: 31.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -2.3 \cdot 10^{-189}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 1.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -2.3e-189)
   (fabs (sin th))
   (if (<= ky 1.2e-236) (/ ky (/ (sin kx) th)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -2.3e-189) {
		tmp = fabs(sin(th));
	} else if (ky <= 1.2e-236) {
		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 <= (-2.3d-189)) then
        tmp = abs(sin(th))
    else if (ky <= 1.2d-236) 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 <= -2.3e-189) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= 1.2e-236) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -2.3e-189:
		tmp = math.fabs(math.sin(th))
	elif ky <= 1.2e-236:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -2.3e-189)
		tmp = abs(sin(th));
	elseif (ky <= 1.2e-236)
		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 <= -2.3e-189)
		tmp = abs(sin(th));
	elseif (ky <= 1.2e-236)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -2.3e-189], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 1.2e-236], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < -2.2999999999999998e-189

   1. Initial program 97.0%

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

      1. associate-*l/ 94.0%

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

      2. associate-*r/ 96.9%

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

      3. +-commutative 96.9%

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

      4. unpow2 96.9%

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

      5. unpow2 96.9%

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

      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. Taylor expanded in kx around 0 21.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. sqrt-unprod 37.1%

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

      3. pow2 37.1%

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

   6. Applied egg-rr 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/r/ 37.2%

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

      3. *-inverses 37.2%

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

      4. /-rgt-identity 37.2%

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

      5. unpow2 37.2%

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

      6. rem-sqrt-square 37.8%

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

   8. Simplified 37.8%

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

   if -2.2999999999999998e-189 < ky < 1.2000000000000001e-236

   1. Initial program 66.7%

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

      1. associate-*l/ 60.3%

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

      2. +-commutative 60.3%

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

      3. unpow2 60.3%

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

      4. unpow2 60.3%

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

      5. hypot-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in th around 0 35.8%

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

   5. Taylor expanded in ky around 0 35.4%

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

      1. associate-/l* 43.6%

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

   7. Simplified 43.6%

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

   if 1.2000000000000001e-236 < ky

   1. Initial program 92.2%

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

      1. associate-*l/ 88.9%

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

      2. associate-*r/ 92.2%

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

      3. +-commutative 92.2%

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

      4. unpow2 92.2%

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

      5. unpow2 92.2%

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

      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. Taylor expanded in kx around 0 37.0%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2.3 \cdot 10^{-189}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 1.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 31.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.9 \cdot 10^{+33}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.4 \cdot 10^{-114}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 7 \cdot 10^{-237}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.9e+33)
   (sin th)
   (if (<= ky -1.4e-114)
     (- th)
     (if (<= ky 7e-237) (/ ky (/ (sin kx) th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.9e+33) {
		tmp = sin(th);
	} else if (ky <= -1.4e-114) {
		tmp = -th;
	} else if (ky <= 7e-237) {
		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 <= (-1.9d+33)) then
        tmp = sin(th)
    else if (ky <= (-1.4d-114)) then
        tmp = -th
    else if (ky <= 7d-237) 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 <= -1.9e+33) {
		tmp = Math.sin(th);
	} else if (ky <= -1.4e-114) {
		tmp = -th;
	} else if (ky <= 7e-237) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.9e+33:
		tmp = math.sin(th)
	elif ky <= -1.4e-114:
		tmp = -th
	elif ky <= 7e-237:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.9e+33)
		tmp = sin(th);
	elseif (ky <= -1.4e-114)
		tmp = Float64(-th);
	elseif (ky <= 7e-237)
		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 <= -1.9e+33)
		tmp = sin(th);
	elseif (ky <= -1.4e-114)
		tmp = -th;
	elseif (ky <= 7e-237)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.9e+33], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -1.4e-114], (-th), If[LessEqual[ky, 7e-237], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.90000000000000001e33 or 6.99999999999999966e-237 < ky

   1. Initial program 95.1%

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

      1. associate-*l/ 93.0%

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

      2. associate-*r/ 95.0%

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

      3. +-commutative 95.0%

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

      4. unpow2 95.0%

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

      5. unpow2 95.0%

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

      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. Taylor expanded in kx around 0 35.9%

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

   if -1.90000000000000001e33 < ky < -1.4000000000000001e-114

   1. Initial program 99.7%

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

      1. associate-*l/ 94.4%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      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. Taylor expanded in kx around 0 3.2%

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

      1. add-sqr-sqrt 1.9%

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

      2. sqrt-unprod 32.2%

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

      3. pow2 32.2%

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

   6. Applied egg-rr 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r/ 32.3%

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

      3. *-inverses 32.3%

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

      4. /-rgt-identity 32.3%

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

   8. Simplified 32.3%

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

   9. Taylor expanded in th around 0 20.2%

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

       1. unpow2 20.2%

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

   11. Simplified 20.2%

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

   12. Taylor expanded in th around -inf 32.6%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 32.6%

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

   14. Simplified 32.6%

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

   if -1.4000000000000001e-114 < ky < 6.99999999999999966e-237

   1. Initial program 71.1%

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

      1. associate-*l/ 63.5%

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

      2. +-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. unpow2 63.5%

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

      5. hypot-def 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in th around 0 36.6%

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

   5. Taylor expanded in ky around 0 33.0%

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

      1. associate-/l* 38.8%

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

   7. Simplified 38.8%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.9 \cdot 10^{+33}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.4 \cdot 10^{-114}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 7 \cdot 10^{-237}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 29.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.9 \cdot 10^{+33}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.85 \cdot 10^{-115}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 3 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.9e+33)
   (sin th)
   (if (<= ky -1.85e-115) (- th) (if (<= ky 3e-251) 0.0 (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.9e+33) {
		tmp = sin(th);
	} else if (ky <= -1.85e-115) {
		tmp = -th;
	} else if (ky <= 3e-251) {
		tmp = 0.0;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.9d+33)) then
        tmp = sin(th)
    else if (ky <= (-1.85d-115)) then
        tmp = -th
    else if (ky <= 3d-251) then
        tmp = 0.0d0
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.9e+33) {
		tmp = Math.sin(th);
	} else if (ky <= -1.85e-115) {
		tmp = -th;
	} else if (ky <= 3e-251) {
		tmp = 0.0;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.9e+33:
		tmp = math.sin(th)
	elif ky <= -1.85e-115:
		tmp = -th
	elif ky <= 3e-251:
		tmp = 0.0
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.9e+33)
		tmp = sin(th);
	elseif (ky <= -1.85e-115)
		tmp = Float64(-th);
	elseif (ky <= 3e-251)
		tmp = 0.0;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.9e+33)
		tmp = sin(th);
	elseif (ky <= -1.85e-115)
		tmp = -th;
	elseif (ky <= 3e-251)
		tmp = 0.0;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.9e+33], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -1.85e-115], (-th), If[LessEqual[ky, 3e-251], 0.0, N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.90000000000000001e33 or 2.9999999999999999e-251 < ky

   1. Initial program 93.5%

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

      1. associate-*l/ 91.4%

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

      2. associate-*r/ 93.4%

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

      3. +-commutative 93.4%

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

      4. unpow2 93.4%

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

      5. unpow2 93.4%

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

      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. Taylor expanded in kx around 0 35.4%

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

   if -1.90000000000000001e33 < ky < -1.85e-115

   1. Initial program 99.7%

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

      1. associate-*l/ 94.4%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      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. Taylor expanded in kx around 0 3.2%

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

      1. add-sqr-sqrt 1.9%

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

      2. sqrt-unprod 32.2%

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

      3. pow2 32.2%

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

   6. Applied egg-rr 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r/ 32.3%

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

      3. *-inverses 32.3%

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

      4. /-rgt-identity 32.3%

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

   8. Simplified 32.3%

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

   9. Taylor expanded in th around 0 20.2%

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

       1. unpow2 20.2%

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

   11. Simplified 20.2%

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

   12. Taylor expanded in th around -inf 32.6%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 32.6%

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

   14. Simplified 32.6%

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

   if -1.85e-115 < ky < 2.9999999999999999e-251

   1. Initial program 74.6%

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

      1. associate-*l/ 66.7%

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

      2. associate-*r/ 74.6%

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

      3. +-commutative 74.6%

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

      4. unpow2 74.6%

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

      5. unpow2 74.6%

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

      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. Taylor expanded in kx around 0 3.9%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 33.2%

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

      3. pow2 33.2%

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

   6. Applied egg-rr 33.2%

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

      1. *-commutative 33.2%

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

      2. associate-/r/ 33.2%

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

      3. *-inverses 33.2%

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

      4. /-rgt-identity 33.2%

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

   8. Simplified 33.2%

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

      1. sqrt-pow1 3.9%

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

      2. metadata-eval 3.9%

         \[\leadsto {\sin th}^{\color{blue}{1}} \]

      3. pow1 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. expm1-udef 28.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

   10. Applied egg-rr 28.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

   11. Taylor expanded in th around 0 30.0%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.9 \cdot 10^{+33}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.85 \cdot 10^{-115}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 3 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 20.4% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -6.2 \cdot 10^{-114}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-244}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -6.2e-114) (- th) (if (<= ky 2.6e-244) 0.0 th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -6.2e-114) {
		tmp = -th;
	} else if (ky <= 2.6e-244) {
		tmp = 0.0;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-6.2d-114)) then
        tmp = -th
    else if (ky <= 2.6d-244) then
        tmp = 0.0d0
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -6.2e-114) {
		tmp = -th;
	} else if (ky <= 2.6e-244) {
		tmp = 0.0;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -6.2e-114:
		tmp = -th
	elif ky <= 2.6e-244:
		tmp = 0.0
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -6.2e-114)
		tmp = Float64(-th);
	elseif (ky <= 2.6e-244)
		tmp = 0.0;
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -6.2e-114)
		tmp = -th;
	elseif (ky <= 2.6e-244)
		tmp = 0.0;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -6.2e-114], (-th), If[LessEqual[ky, 2.6e-244], 0.0, th]]

Derivation

1. Split input into 3 regimes

2. if ky < -6.2e-114

   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/ 98.2%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      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. Taylor expanded in kx around 0 25.4%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-unprod 34.7%

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

      3. pow2 34.7%

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

   6. Applied egg-rr 34.7%

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

      1. *-commutative 34.7%

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

      2. associate-/r/ 34.8%

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

      3. *-inverses 34.8%

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

      4. /-rgt-identity 34.8%

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

   8. Simplified 34.8%

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

   9. Taylor expanded in th around 0 14.0%

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

       1. unpow2 14.0%

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

   11. Simplified 14.0%

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

   12. Taylor expanded in th around -inf 25.9%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 25.9%

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

   14. Simplified 25.9%

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

   if -6.2e-114 < ky < 2.6000000000000001e-244

   1. Initial program 72.2%

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

      1. associate-*l/ 64.5%

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

      2. associate-*r/ 72.2%

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

      3. +-commutative 72.2%

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

      4. unpow2 72.2%

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

      5. unpow2 72.2%

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

      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. Taylor expanded in kx around 0 4.1%

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

      1. add-sqr-sqrt 2.5%

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

      2. sqrt-unprod 32.4%

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

      3. pow2 32.4%

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

   6. Applied egg-rr 32.4%

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

      1. *-commutative 32.4%

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

      2. associate-/r/ 32.4%

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

      3. *-inverses 32.4%

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

      4. /-rgt-identity 32.4%

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

   8. Simplified 32.4%

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

      1. sqrt-pow1 4.1%

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

      2. metadata-eval 4.1%

         \[\leadsto {\sin th}^{\color{blue}{1}} \]

      3. pow1 4.1%

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

      4. expm1-log1p-u 4.1%

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

      5. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

   10. Applied egg-rr 28.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

   11. Taylor expanded in th around 0 29.1%

       \[\leadsto \color{blue}{1} - 1 \]

   if 2.6000000000000001e-244 < ky

   1. Initial program 91.4%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 91.3%

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

      3. +-commutative 91.3%

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

      4. unpow2 91.3%

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

      5. unpow2 91.3%

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

      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. Taylor expanded in kx around 0 36.6%

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

   5. Taylor expanded in th around 0 23.4%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.2 \cdot 10^{-114}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-244}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 16: 16.1% accurate, 174.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 8 \cdot 10^{-246}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= ky 8e-246) (- th) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8e-246) {
		tmp = -th;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 8d-246) then
        tmp = -th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8e-246) {
		tmp = -th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 8e-246:
		tmp = -th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 8e-246)
		tmp = Float64(-th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 8e-246)
		tmp = -th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 8e-246], (-th), th]

Derivation

1. Split input into 2 regimes

2. if ky < 7.99999999999999965e-246

   1. Initial program 88.7%

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

      1. associate-*l/ 84.7%

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

      2. associate-*r/ 88.6%

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

      3. +-commutative 88.6%

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

      4. unpow2 88.6%

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

      5. unpow2 88.6%

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

      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. Taylor expanded in kx around 0 16.9%

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

      1. add-sqr-sqrt 9.3%

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

      2. sqrt-unprod 33.8%

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

      3. pow2 33.8%

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

   6. Applied egg-rr 33.8%

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

      1. *-commutative 33.8%

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

      2. associate-/r/ 33.8%

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

      3. *-inverses 33.8%

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

      4. /-rgt-identity 33.8%

         \[\leadsto \sqrt{{\color{blue}{\sin th}}^{2}} \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]

   9. Taylor expanded in th around 0 17.0%

      \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 17.0%

          \[\leadsto \sqrt{\color{blue}{th \cdot th}} \]

   11. Simplified 17.0%

       \[\leadsto \sqrt{\color{blue}{th \cdot th}} \]

   12. Taylor expanded in th around -inf 17.9%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   13. Step-by-step derivation

       1. neg-mul-1 17.9%

          \[\leadsto \color{blue}{-th} \]

   14. Simplified 17.9%

       \[\leadsto \color{blue}{-th} \]

   if 7.99999999999999965e-246 < ky

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 88.1%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-*r/ 91.3%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. +-commutative 91.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      4. unpow2 91.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      5. unpow2 91.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      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. Taylor expanded in kx around 0 36.6%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

   5. Taylor expanded in th around 0 23.4%

      \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 8 \cdot 10^{-246}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 17: 13.0% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 89.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-*l/ 86.1%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. associate-*r/ 89.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. +-commutative 89.7%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

   4. unpow2 89.7%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

   5. unpow2 89.7%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

   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. Taylor expanded in kx around 0 25.1%

   \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

5. Taylor expanded in th around 0 15.3%

   \[\leadsto \color{blue}{th} \]

6. Final simplification 15.3%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023175 
(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)))