Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.6%

Time: 21.3s

Alternatives: 16

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: 94.2% 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: 99.6% 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 91.6%

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

   1. associate-*l/ 89.9%

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

   2. associate-*r/ 91.6%

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

   3. +-commutative 91.6%

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

   4. unpow2 91.6%

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

   5. unpow2 91.6%

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

   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. Final simplification 99.7%

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

Alternative 2: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{if}\;\sin th \leq -0.8:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.65:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq -0.01:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.87:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) (sin th)) (fabs (sin kx)))))
   (if (<= (sin th) -0.8)
     t_1
     (if (<= (sin th) -0.65)
       (fabs (sin th))
       (if (<= (sin th) -0.01)
         t_1
         (if (<= (sin th) 1e-7)
           (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
           (if (<= (sin th) 0.87)
             (/ (sin ky) (fabs (/ (sin ky) (sin th))))
             (* (sin ky) (fabs (/ (sin th) (sin kx)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * sin(th)) / fabs(sin(kx));
	double tmp;
	if (sin(th) <= -0.8) {
		tmp = t_1;
	} else if (sin(th) <= -0.65) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= -0.01) {
		tmp = t_1;
	} else if (sin(th) <= 1e-7) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(th) <= 0.87) {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * Math.sin(th)) / Math.abs(Math.sin(kx));
	double tmp;
	if (Math.sin(th) <= -0.8) {
		tmp = t_1;
	} else if (Math.sin(th) <= -0.65) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= -0.01) {
		tmp = t_1;
	} else if (Math.sin(th) <= 1e-7) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(th) <= 0.87) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * math.sin(th)) / math.fabs(math.sin(kx))
	tmp = 0
	if math.sin(th) <= -0.8:
		tmp = t_1
	elif math.sin(th) <= -0.65:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= -0.01:
		tmp = t_1
	elif math.sin(th) <= 1e-7:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(th) <= 0.87:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * sin(th)) / abs(sin(kx)))
	tmp = 0.0
	if (sin(th) <= -0.8)
		tmp = t_1;
	elseif (sin(th) <= -0.65)
		tmp = abs(sin(th));
	elseif (sin(th) <= -0.01)
		tmp = t_1;
	elseif (sin(th) <= 1e-7)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(th) <= 0.87)
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * sin(th)) / abs(sin(kx));
	tmp = 0.0;
	if (sin(th) <= -0.8)
		tmp = t_1;
	elseif (sin(th) <= -0.65)
		tmp = abs(sin(th));
	elseif (sin(th) <= -0.01)
		tmp = t_1;
	elseif (sin(th) <= 1e-7)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(th) <= 0.87)
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.8], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], -0.65], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.01], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], 1e-7], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.87], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 th) < -0.80000000000000004 or -0.650000000000000022 < (sin.f64 th) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. associate-*l/ 93.8%

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

   3. Simplified 93.8%

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

      1. unpow2 93.8%

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

      2. sin-mult 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. div-sub 86.5%

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

      2. +-inverses 86.5%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \]

      3. cos-0 86.5%

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

      4. metadata-eval 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in ky around 0 36.8%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 36.8%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \]

      2. rem-sqrt-square 36.8%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \]

      3. sqr-sin-a 44.6%

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

      4. sqrt-unprod 17.4%

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

      5. add-sqr-sqrt 45.2%

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

   10. Applied egg-rr 45.2%

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

   if -0.80000000000000004 < (sin.f64 th) < -0.650000000000000022

   1. Initial program 85.7%

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

      1. associate-/r/ 85.9%

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

      2. +-commutative 85.9%

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

      3. unpow2 85.9%

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

      4. unpow2 85.9%

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

      5. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 4.3%

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

      1. associate-/r/ 4.3%

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

      2. *-inverses 4.3%

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

      3. *-un-lft-identity 4.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 74.5%

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

      6. pow2 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. unpow2 74.5%

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

      2. rem-sqrt-square 74.5%

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

   8. Simplified 74.5%

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

   if -0.0100000000000000002 < (sin.f64 th) < 9.9999999999999995e-8

   1. Initial program 91.2%

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

      1. associate-*l/ 87.9%

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

      2. +-commutative 87.9%

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

      3. unpow2 87.9%

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

      4. unpow2 87.9%

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

      5. hypot-def 91.4%

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

   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 91.1%

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

      1. expm1-log1p-u 91.1%

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

      2. expm1-udef 23.1%

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

   6. Applied egg-rr 23.1%

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

      1. expm1-def 91.1%

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

      2. expm1-log1p 91.1%

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

      3. associate-*r/ 99.5%

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

   8. Simplified 99.5%

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

   if 9.9999999999999995e-8 < (sin.f64 th) < 0.869999999999999996

   1. Initial program 87.1%

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

      1. associate-/r/ 87.1%

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

      2. +-commutative 87.1%

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

      3. unpow2 87.1%

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

      4. unpow2 87.1%

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

      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 kx around 0 32.1%

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

      1. add-sqr-sqrt 31.1%

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

      2. sqrt-unprod 51.8%

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

      3. pow2 51.8%

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

   6. Applied egg-rr 51.8%

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

      1. unpow2 51.8%

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

      2. rem-sqrt-square 58.9%

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

   8. Simplified 58.9%

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

   if 0.869999999999999996 < (sin.f64 th)

   1. Initial program 99.4%

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

      1. associate-*l/ 99.6%

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

      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. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 36.7%

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

      1. add-sqr-sqrt 35.2%

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

      2. sqrt-unprod 60.4%

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

      3. pow2 60.4%

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

      4. associate-*l/ 60.7%

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

      5. *-un-lft-identity 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. unpow2 60.7%

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

      2. rem-sqrt-square 60.8%

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

   10. Simplified 60.8%

       \[\leadsto \sin ky \cdot \color{blue}{\left|\frac{\sin th}{\sin kx}\right|} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.8:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin th \leq -0.65:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq -0.01:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.87:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]

Alternative 3: 49.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.85:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.85)
   (fabs (sin th))
   (if (<= (sin ky) -0.005)
     (/ (sin ky) (fabs (/ (sin ky) (sin th))))
     (if (<= (sin ky) -2e-158)
       (/ (* ky (sin th)) (sqrt (- 0.5 (* 0.5 (cos (* kx 2.0))))))
       (if (<= (sin ky) 5e-128)
         (fabs (* (sin th) (/ (sin ky) (sin kx))))
         (if (<= (sin ky) 5e-6)
           (/ (* ky th) (hypot (sin ky) (sin kx)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.85) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -0.005) {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	} else if (sin(ky) <= -2e-158) {
		tmp = (ky * sin(th)) / sqrt((0.5 - (0.5 * cos((kx * 2.0)))));
	} else if (sin(ky) <= 5e-128) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else if (sin(ky) <= 5e-6) {
		tmp = (ky * th) / hypot(sin(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) <= -0.85) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -0.005) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	} else if (Math.sin(ky) <= -2e-158) {
		tmp = (ky * Math.sin(th)) / Math.sqrt((0.5 - (0.5 * Math.cos((kx * 2.0)))));
	} else if (Math.sin(ky) <= 5e-128) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else if (Math.sin(ky) <= 5e-6) {
		tmp = (ky * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.85:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -0.005:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	elif math.sin(ky) <= -2e-158:
		tmp = (ky * math.sin(th)) / math.sqrt((0.5 - (0.5 * math.cos((kx * 2.0)))))
	elif math.sin(ky) <= 5e-128:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	elif math.sin(ky) <= 5e-6:
		tmp = (ky * th) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.85)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.005)
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	elseif (sin(ky) <= -2e-158)
		tmp = Float64(Float64(ky * sin(th)) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0))))));
	elseif (sin(ky) <= 5e-128)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	elseif (sin(ky) <= 5e-6)
		tmp = Float64(Float64(ky * th) / hypot(sin(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) <= -0.85)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.005)
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	elseif (sin(ky) <= -2e-158)
		tmp = (ky * sin(th)) / sqrt((0.5 - (0.5 * cos((kx * 2.0)))));
	elseif (sin(ky) <= 5e-128)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	elseif (sin(ky) <= 5e-6)
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.85], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-158], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-128], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-6], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (sin.f64 ky) < -0.849999999999999978

   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-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      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 kx around 0 2.6%

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

      1. associate-/r/ 2.6%

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

      2. *-inverses 2.6%

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

      3. *-un-lft-identity 2.6%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 39.0%

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

      6. pow2 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. unpow2 39.0%

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

      2. rem-sqrt-square 45.6%

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

   8. Simplified 45.6%

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

   if -0.849999999999999978 < (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 26.8%

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

      3. pow2 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. unpow2 26.8%

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

      2. rem-sqrt-square 36.0%

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

   8. Simplified 36.0%

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

   if -0.0050000000000000001 < (sin.f64 ky) < -2.00000000000000013e-158

   1. Initial program 99.5%

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

      1. associate-*l/ 96.4%

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

   3. Simplified 96.4%

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

      1. unpow2 96.4%

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

      2. sin-mult 89.3%

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

   5. Applied egg-rr 89.3%

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

      1. div-sub 89.3%

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

      2. +-inverses 89.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \]

      3. cos-0 89.3%

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

      4. metadata-eval 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in ky around 0 56.5%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   9. Taylor expanded in ky around 0 56.5%

      \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

   if -2.00000000000000013e-158 < (sin.f64 ky) < 5.0000000000000001e-128

   1. Initial program 72.4%

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

      1. associate-*l/ 69.3%

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

      2. associate-*r/ 72.4%

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

      3. +-commutative 72.4%

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

      4. unpow2 72.4%

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

      5. unpow2 72.4%

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

      6. hypot-def 99.7%

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

   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. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in ky around 0 58.2%

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

      1. add-sqr-sqrt 47.4%

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

      2. sqrt-unprod 43.4%

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

      3. pow2 43.4%

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

      4. associate-*l/ 43.4%

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

      5. *-un-lft-identity 43.4%

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

   8. Applied egg-rr 43.4%

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

      1. unpow2 43.4%

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

      2. rem-sqrt-square 70.7%

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

      3. associate-*r/ 66.8%

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

      4. associate-*l/ 70.6%

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

      5. *-commutative 70.6%

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

   10. Simplified 70.6%

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

   if 5.0000000000000001e-128 < (sin.f64 ky) < 5.00000000000000041e-6

   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/ 95.5%

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

      2. +-commutative 95.5%

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

      3. unpow2 95.5%

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

      4. unpow2 95.5%

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

      5. hypot-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in th around 0 63.5%

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

   5. Taylor expanded in ky around 0 61.9%

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

   if 5.00000000000000041e-6 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 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 67.4%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.85:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.85:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -0.4:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -0.15:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 10^{-42}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin ky) -0.85)
     t_1
     (if (<= (sin ky) -0.4)
       (/ (sin ky) (fabs (/ (sin ky) (sin th))))
       (if (<= (sin ky) -0.15)
         t_1
         (if (<= (sin ky) 1e-42)
           (/ (* (sin ky) (sin th)) (fabs (sin kx)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(ky) <= -0.85) {
		tmp = t_1;
	} else if (sin(ky) <= -0.4) {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	} else if (sin(ky) <= -0.15) {
		tmp = t_1;
	} else if (sin(ky) <= 1e-42) {
		tmp = (sin(ky) * sin(th)) / fabs(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) <= (-0.85d0)) then
        tmp = t_1
    else if (sin(ky) <= (-0.4d0)) then
        tmp = sin(ky) / abs((sin(ky) / sin(th)))
    else if (sin(ky) <= (-0.15d0)) then
        tmp = t_1
    else if (sin(ky) <= 1d-42) then
        tmp = (sin(ky) * sin(th)) / abs(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) <= -0.85) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -0.4) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	} else if (Math.sin(ky) <= -0.15) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 1e-42) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.abs(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) <= -0.85:
		tmp = t_1
	elif math.sin(ky) <= -0.4:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	elif math.sin(ky) <= -0.15:
		tmp = t_1
	elif math.sin(ky) <= 1e-42:
		tmp = (math.sin(ky) * math.sin(th)) / math.fabs(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) <= -0.85)
		tmp = t_1;
	elseif (sin(ky) <= -0.4)
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	elseif (sin(ky) <= -0.15)
		tmp = t_1;
	elseif (sin(ky) <= 1e-42)
		tmp = Float64(Float64(sin(ky) * sin(th)) / abs(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) <= -0.85)
		tmp = t_1;
	elseif (sin(ky) <= -0.4)
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	elseif (sin(ky) <= -0.15)
		tmp = t_1;
	elseif (sin(ky) <= 1e-42)
		tmp = (sin(ky) * sin(th)) / abs(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], -0.85], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.4], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.15], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-42], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.849999999999999978 or -0.40000000000000002 < (sin.f64 ky) < -0.149999999999999994

   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-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      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 kx around 0 2.7%

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

      1. associate-/r/ 2.7%

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

      2. *-inverses 2.7%

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

      3. *-un-lft-identity 2.7%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 40.1%

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

      6. pow2 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. unpow2 40.1%

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

      2. rem-sqrt-square 48.3%

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

   8. Simplified 48.3%

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

   if -0.849999999999999978 < (sin.f64 ky) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. associate-/r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 29.2%

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

      3. pow2 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 37.7%

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

   8. Simplified 37.7%

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

   if -0.149999999999999994 < (sin.f64 ky) < 1.00000000000000004e-42

   1. Initial program 83.2%

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

      1. associate-*l/ 80.6%

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

   3. Simplified 80.6%

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

      1. unpow2 80.6%

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

      2. sin-mult 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. div-sub 64.4%

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

      2. +-inverses 64.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \]

      3. cos-0 64.4%

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

      4. metadata-eval 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in ky around 0 50.1%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 50.1%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \]

      2. rem-sqrt-square 50.1%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \]

      3. sqr-sin-a 66.8%

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

      4. sqrt-unprod 37.9%

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

      5. add-sqr-sqrt 71.6%

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

   10. Applied egg-rr 71.6%

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

   if 1.00000000000000004e-42 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 98.4%

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

      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 66.7%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.85:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.4:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-42}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.15)
   (fabs (sin th))
   (if (<= (sin ky) -2e-158)
     (/ (* ky (sin th)) (sqrt (- 0.5 (* 0.5 (cos (* kx 2.0))))))
     (if (<= (sin ky) 5e-128)
       (fabs (* (sin th) (/ (sin ky) (sin kx))))
       (if (<= (sin ky) 5e-6)
         (/ (* ky th) (hypot (sin ky) (sin kx)))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.15) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -2e-158) {
		tmp = (ky * sin(th)) / sqrt((0.5 - (0.5 * cos((kx * 2.0)))));
	} else if (sin(ky) <= 5e-128) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else if (sin(ky) <= 5e-6) {
		tmp = (ky * th) / hypot(sin(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) <= -0.15) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -2e-158) {
		tmp = (ky * Math.sin(th)) / Math.sqrt((0.5 - (0.5 * Math.cos((kx * 2.0)))));
	} else if (Math.sin(ky) <= 5e-128) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else if (Math.sin(ky) <= 5e-6) {
		tmp = (ky * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.15:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -2e-158:
		tmp = (ky * math.sin(th)) / math.sqrt((0.5 - (0.5 * math.cos((kx * 2.0)))))
	elif math.sin(ky) <= 5e-128:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	elif math.sin(ky) <= 5e-6:
		tmp = (ky * th) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.15)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -2e-158)
		tmp = Float64(Float64(ky * sin(th)) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0))))));
	elseif (sin(ky) <= 5e-128)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	elseif (sin(ky) <= 5e-6)
		tmp = Float64(Float64(ky * th) / hypot(sin(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) <= -0.15)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -2e-158)
		tmp = (ky * sin(th)) / sqrt((0.5 - (0.5 * cos((kx * 2.0)))));
	elseif (sin(ky) <= 5e-128)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	elseif (sin(ky) <= 5e-6)
		tmp = (ky * th) / hypot(sin(ky), sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.15], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-158], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-128], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-6], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.149999999999999994

   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-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 1.5%

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

      5. sqrt-unprod 29.2%

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

      6. pow2 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 35.7%

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

   8. Simplified 35.7%

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

   if -0.149999999999999994 < (sin.f64 ky) < -2.00000000000000013e-158

   1. Initial program 99.5%

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

      1. associate-*l/ 96.7%

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

   3. Simplified 96.7%

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

      1. unpow2 96.7%

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

      2. sin-mult 90.3%

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

   5. Applied egg-rr 90.3%

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

      1. div-sub 90.3%

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

      2. +-inverses 90.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \]

      3. cos-0 90.3%

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

      4. metadata-eval 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in ky around 0 53.1%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   9. Taylor expanded in ky around 0 51.6%

      \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

   if -2.00000000000000013e-158 < (sin.f64 ky) < 5.0000000000000001e-128

   1. Initial program 72.4%

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

      1. associate-*l/ 69.3%

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

      2. associate-*r/ 72.4%

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

      3. +-commutative 72.4%

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

      4. unpow2 72.4%

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

      5. unpow2 72.4%

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

      6. hypot-def 99.7%

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

   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. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in ky around 0 58.2%

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

      1. add-sqr-sqrt 47.4%

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

      2. sqrt-unprod 43.4%

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

      3. pow2 43.4%

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

      4. associate-*l/ 43.4%

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

      5. *-un-lft-identity 43.4%

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

   8. Applied egg-rr 43.4%

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

      1. unpow2 43.4%

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

      2. rem-sqrt-square 70.7%

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

      3. associate-*r/ 66.8%

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

      4. associate-*l/ 70.6%

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

      5. *-commutative 70.6%

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

   10. Simplified 70.6%

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

   if 5.0000000000000001e-128 < (sin.f64 ky) < 5.00000000000000041e-6

   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/ 95.5%

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

      2. +-commutative 95.5%

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

      3. unpow2 95.5%

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

      4. unpow2 95.5%

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

      5. hypot-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in th around 0 63.5%

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

   5. Taylor expanded in ky around 0 61.9%

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

   if 5.00000000000000041e-6 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 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 67.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin th \leq -0.01:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin th \leq 0.87:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin th) -0.01)
     (/ (* ky (sin th)) t_1)
     (if (<= (sin th) 1e-7)
       (* (sin ky) (/ th t_1))
       (if (<= (sin th) 0.87)
         (/ (sin ky) (fabs (/ (sin ky) (sin th))))
         (* (sin ky) (fabs (/ (sin th) (sin kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(th) <= -0.01) {
		tmp = (ky * sin(th)) / t_1;
	} else if (sin(th) <= 1e-7) {
		tmp = sin(ky) * (th / t_1);
	} else if (sin(th) <= 0.87) {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(th) <= -0.01) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else if (Math.sin(th) <= 1e-7) {
		tmp = Math.sin(ky) * (th / t_1);
	} else if (Math.sin(th) <= 0.87) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(th) <= -0.01:
		tmp = (ky * math.sin(th)) / t_1
	elif math.sin(th) <= 1e-7:
		tmp = math.sin(ky) * (th / t_1)
	elif math.sin(th) <= 0.87:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(th) <= -0.01)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	elseif (sin(th) <= 1e-7)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	elseif (sin(th) <= 0.87)
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(th) <= -0.01)
		tmp = (ky * sin(th)) / t_1;
	elseif (sin(th) <= 1e-7)
		tmp = sin(ky) * (th / t_1);
	elseif (sin(th) <= 0.87)
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.01], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-7], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.87], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 92.8%

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

      1. associate-*l/ 92.9%

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

      2. +-commutative 92.9%

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

      3. unpow2 92.9%

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

      4. unpow2 92.9%

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

      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 49.8%

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

   if -0.0100000000000000002 < (sin.f64 th) < 9.9999999999999995e-8

   1. Initial program 91.2%

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

      1. associate-*l/ 87.9%

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

      2. +-commutative 87.9%

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

      3. unpow2 87.9%

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

      4. unpow2 87.9%

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

      5. hypot-def 91.4%

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

   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 91.1%

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

      1. expm1-log1p-u 91.1%

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

      2. expm1-udef 23.1%

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

   6. Applied egg-rr 23.1%

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

      1. expm1-def 91.1%

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

      2. expm1-log1p 91.1%

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

      3. associate-*r/ 99.5%

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

   8. Simplified 99.5%

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

   if 9.9999999999999995e-8 < (sin.f64 th) < 0.869999999999999996

   1. Initial program 87.1%

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

      1. associate-/r/ 87.1%

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

      2. +-commutative 87.1%

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

      3. unpow2 87.1%

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

      4. unpow2 87.1%

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

      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 kx around 0 32.1%

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

      1. add-sqr-sqrt 31.1%

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

      2. sqrt-unprod 51.8%

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

      3. pow2 51.8%

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

   6. Applied egg-rr 51.8%

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

      1. unpow2 51.8%

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

      2. rem-sqrt-square 58.9%

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

   8. Simplified 58.9%

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

   if 0.869999999999999996 < (sin.f64 th)

   1. Initial program 99.4%

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

      1. associate-*l/ 99.6%

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

      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. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 36.7%

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

      1. add-sqr-sqrt 35.2%

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

      2. sqrt-unprod 60.4%

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

      3. pow2 60.4%

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

      4. associate-*l/ 60.7%

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

      5. *-un-lft-identity 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. unpow2 60.7%

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

      2. rem-sqrt-square 60.8%

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

   10. Simplified 60.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.01:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.87:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]

Alternative 7: 47.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-42}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.15)
   (fabs (sin th))
   (if (<= (sin ky) 1e-42) (* (sin ky) (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.15) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-42) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.15d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-42) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.15) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-42) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.15:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-42:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.15)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-42)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.15)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-42)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.15], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-42], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.149999999999999994

   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-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 1.5%

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

      5. sqrt-unprod 29.2%

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

      6. pow2 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 35.7%

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

   8. Simplified 35.7%

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

   if -0.149999999999999994 < (sin.f64 ky) < 1.00000000000000004e-42

   1. Initial program 83.2%

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

      1. associate-*l/ 80.6%

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

      2. associate-*r/ 83.2%

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

      3. +-commutative 83.2%

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

      4. unpow2 83.2%

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

      5. unpow2 83.2%

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

      6. hypot-def 99.7%

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

   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 ky around 0 49.1%

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

   if 1.00000000000000004e-42 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 98.4%

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

      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 66.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-42}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 49.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (/ (* ky (sin th)) (sqrt (- 0.5 (* 0.5 (cos (* kx 2.0))))))
   (if (<= (sin kx) 2e-89)
     (fabs (sin th))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = (ky * sin(th)) / sqrt((0.5 - (0.5 * cos((kx * 2.0)))));
	} else if (sin(kx) <= 2e-89) {
		tmp = fabs(sin(th));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = (ky * sin(th)) / sqrt((0.5d0 - (0.5d0 * cos((kx * 2.0d0)))))
    else if (sin(kx) <= 2d-89) then
        tmp = abs(sin(th))
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = (ky * Math.sin(th)) / Math.sqrt((0.5 - (0.5 * Math.cos((kx * 2.0)))));
	} else if (Math.sin(kx) <= 2e-89) {
		tmp = Math.abs(Math.sin(th));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = (ky * math.sin(th)) / math.sqrt((0.5 - (0.5 * math.cos((kx * 2.0)))))
	elif math.sin(kx) <= 2e-89:
		tmp = math.fabs(math.sin(th))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = Float64(Float64(ky * sin(th)) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0))))));
	elseif (sin(kx) <= 2e-89)
		tmp = abs(sin(th));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = (ky * sin(th)) / sqrt((0.5 - (0.5 * cos((kx * 2.0)))));
	elseif (sin(kx) <= 2e-89)
		tmp = abs(sin(th));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-89], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.2%

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

      1. associate-*l/ 99.5%

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

   3. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. div-sub 99.1%

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

      2. +-inverses 99.1%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \]

      3. cos-0 99.1%

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

      4. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in ky around 0 62.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   9. Taylor expanded in ky around 0 54.5%

      \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

   if -0.0100000000000000002 < (sin.f64 kx) < 2.00000000000000008e-89

   1. Initial program 81.4%

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

      1. associate-/r/ 81.3%

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

      2. +-commutative 81.3%

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

      3. unpow2 81.3%

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

      4. unpow2 81.3%

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

      5. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 42.1%

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

      1. associate-/r/ 42.1%

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

      2. *-inverses 42.1%

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

      3. *-un-lft-identity 42.1%

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

      4. add-sqr-sqrt 24.9%

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

      5. sqrt-unprod 44.0%

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

      6. pow2 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. unpow2 44.0%

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

      2. rem-sqrt-square 49.3%

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

   8. Simplified 49.3%

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

   if 2.00000000000000008e-89 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. associate-*l/ 99.7%

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

      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 ky around 0 60.2%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 9: 40.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-137}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-137)
   (fabs (sin th))
   (if (<= (sin ky) 2e-191) (* (sin ky) (/ th (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-137) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-191) {
		tmp = sin(ky) * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-5d-137)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-191) then
        tmp = sin(ky) * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -5e-137) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-191) {
		tmp = Math.sin(ky) * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-137:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-191:
		tmp = math.sin(ky) * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-137)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-191)
		tmp = Float64(sin(ky) * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-137)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-191)
		tmp = sin(ky) * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-137], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-191], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.00000000000000001e-137

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      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 kx around 0 2.9%

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

      1. associate-/r/ 2.9%

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

      2. *-inverses 2.9%

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

      3. *-un-lft-identity 2.9%

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

      4. add-sqr-sqrt 1.4%

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

      5. sqrt-unprod 27.7%

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

      6. pow2 27.7%

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

   6. Applied egg-rr 27.7%

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

      1. unpow2 27.7%

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

      2. rem-sqrt-square 33.0%

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

   8. Simplified 33.0%

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

   if -5.00000000000000001e-137 < (sin.f64 ky) < 2e-191

   1. Initial program 70.7%

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

      1. associate-*l/ 67.1%

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

      2. associate-*r/ 70.8%

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

      3. +-commutative 70.8%

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

      4. unpow2 70.8%

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

      5. unpow2 70.8%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in ky around 0 57.8%

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

   7. Taylor expanded in th around 0 41.5%

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

   if 2e-191 < (sin.f64 ky)

   1. Initial program 97.5%

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

      1. associate-*l/ 96.6%

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

      2. associate-*r/ 97.4%

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

      3. +-commutative 97.4%

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

      4. unpow2 97.4%

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

      5. unpow2 97.4%

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

      6. hypot-def 99.7%

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

   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 54.3%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-137}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 42.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-97}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-134}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -4e-97)
   (fabs (sin th))
   (if (<= (sin ky) 1e-134) (* (sin ky) (/ (sin th) kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -4e-97) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-134) {
		tmp = sin(ky) * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-4d-97)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-134) then
        tmp = sin(ky) * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -4e-97) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-134) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -4e-97:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-134:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -4e-97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-134)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -4e-97)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-134)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -4e-97], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-134], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -4.00000000000000014e-97

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      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 kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 1.4%

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

      5. sqrt-unprod 26.0%

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

      6. pow2 26.0%

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

   6. Applied egg-rr 26.0%

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

      1. unpow2 26.0%

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

      2. rem-sqrt-square 33.0%

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

   8. Simplified 33.0%

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

   if -4.00000000000000014e-97 < (sin.f64 ky) < 1.00000000000000004e-134

   1. Initial program 75.5%

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

      1. associate-*l/ 71.8%

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

      2. associate-*r/ 75.5%

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

      3. +-commutative 75.5%

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

      4. unpow2 75.5%

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

      5. unpow2 75.5%

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

      6. hypot-def 99.7%

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

   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. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in ky around 0 55.2%

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

   7. Taylor expanded in kx around 0 43.7%

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

   if 1.00000000000000004e-134 < (sin.f64 ky)

   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/ 98.7%

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

      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 58.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-97}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-134}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-42}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (fabs (sin th))
   (if (<= (sin ky) 1e-42) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-42) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-42) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-42) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-42:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-42)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-42)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-42], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      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 kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 1.5%

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

      5. sqrt-unprod 27.9%

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

      6. pow2 27.9%

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

   6. Applied egg-rr 27.9%

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

      1. unpow2 27.9%

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

      2. rem-sqrt-square 34.2%

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

   8. Simplified 34.2%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 1.00000000000000004e-42

   1. Initial program 82.8%

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

      1. associate-*l/ 80.1%

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

      2. associate-*r/ 82.8%

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

      3. +-commutative 82.8%

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

      4. unpow2 82.8%

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

      5. unpow2 82.8%

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

      6. hypot-def 99.7%

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

   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 ky around 0 47.1%

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

      1. associate-/l* 49.8%

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

   6. Simplified 49.8%

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

   if 1.00000000000000004e-42 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 98.4%

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

      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 66.7%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-42}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 31.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{-137} \lor \neg \left(ky \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -1.1e-137) (not (<= ky 3.7e-6)))
   (fabs (sin th))
   (/ ky (/ (sin kx) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.1e-137) || !(ky <= 3.7e-6)) {
		tmp = fabs(sin(th));
	} else {
		tmp = ky / (sin(kx) / 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 <= (-1.1d-137)) .or. (.not. (ky <= 3.7d-6))) then
        tmp = abs(sin(th))
    else
        tmp = ky / (sin(kx) / th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.1e-137) || !(ky <= 3.7e-6)) {
		tmp = Math.abs(Math.sin(th));
	} else {
		tmp = ky / (Math.sin(kx) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -1.1e-137) or not (ky <= 3.7e-6):
		tmp = math.fabs(math.sin(th))
	else:
		tmp = ky / (math.sin(kx) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -1.1e-137) || !(ky <= 3.7e-6))
		tmp = abs(sin(th));
	else
		tmp = Float64(ky / Float64(sin(kx) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -1.1e-137) || ~((ky <= 3.7e-6)))
		tmp = abs(sin(th));
	else
		tmp = ky / (sin(kx) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -1.1e-137], N[Not[LessEqual[ky, 3.7e-6]], $MachinePrecision]], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.1000000000000001e-137 or 3.7000000000000002e-6 < ky

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      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 kx around 0 29.5%

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

      1. associate-/r/ 29.5%

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

      2. *-inverses 29.5%

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

      3. *-un-lft-identity 29.5%

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

      4. add-sqr-sqrt 18.9%

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

      5. sqrt-unprod 32.1%

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

      6. pow2 32.1%

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

   6. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. rem-sqrt-square 38.1%

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

   8. Simplified 38.1%

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

   if -1.1000000000000001e-137 < ky < 3.7000000000000002e-6

   1. Initial program 79.1%

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

      1. associate-*l/ 76.0%

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

      2. +-commutative 76.0%

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

      3. unpow2 76.0%

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

      4. unpow2 76.0%

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

      5. hypot-def 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in th around 0 44.2%

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

   5. Taylor expanded in ky around 0 33.4%

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

      1. associate-/l* 36.6%

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

   7. Simplified 36.6%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{-137} \lor \neg \left(ky \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \]

Alternative 13: 32.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -22:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -22.0)
   (sin th)
   (if (<= ky 1.8e-189) (/ ky (/ (sin kx) th)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -22.0) {
		tmp = sin(th);
	} else if (ky <= 1.8e-189) {
		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 <= (-22.0d0)) then
        tmp = sin(th)
    else if (ky <= 1.8d-189) 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 <= -22.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1.8e-189) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -22.0:
		tmp = math.sin(th)
	elif ky <= 1.8e-189:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -22.0)
		tmp = sin(th);
	elseif (ky <= 1.8e-189)
		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 <= -22.0)
		tmp = sin(th);
	elseif (ky <= 1.8e-189)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -22.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.8e-189], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -22 or 1.80000000000000008e-189 < ky

   1. Initial program 98.3%

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

      1. associate-*l/ 97.8%

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

      2. associate-*r/ 98.3%

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

      3. +-commutative 98.3%

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

      4. unpow2 98.3%

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

      5. unpow2 98.3%

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

      6. hypot-def 99.7%

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

   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 34.1%

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

   if -22 < ky < 1.80000000000000008e-189

   1. Initial program 79.0%

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

      1. associate-*l/ 75.4%

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

      2. +-commutative 75.4%

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

      3. unpow2 75.4%

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

      4. unpow2 75.4%

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

      5. hypot-def 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in th around 0 42.5%

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

   5. Taylor expanded in ky around 0 31.3%

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

      1. associate-/l* 35.0%

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

   7. Simplified 35.0%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -22:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 25.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.65 \cdot 10^{-12}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 1.65e-12) (sin th) 0.0))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.65e-12) {
		tmp = sin(th);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.65d-12) then
        tmp = sin(th)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.65e-12) {
		tmp = Math.sin(th);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.65e-12:
		tmp = math.sin(th)
	else:
		tmp = 0.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.65e-12)
		tmp = sin(th);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.65e-12)
		tmp = sin(th);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.65e-12], N[Sin[th], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if kx < 1.65e-12

   1. Initial program 88.8%

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

      1. associate-*l/ 86.6%

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

      2. associate-*r/ 88.8%

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

      3. +-commutative 88.8%

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

      4. unpow2 88.8%

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

      5. unpow2 88.8%

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

      6. hypot-def 99.7%

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

   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 30.2%

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

   if 1.65e-12 < kx

   1. Initial program 99.4%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 7.2%

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

      1. associate-/r/ 7.2%

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

      2. *-inverses 7.2%

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

      3. *-un-lft-identity 7.2%

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

      4. add-log-exp 19.5%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   6. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   7. Taylor expanded in th around 0 17.8%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.65 \cdot 10^{-12}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 16.3% accurate, 231.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 1.45e-13) th 0.0))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e-13) {
		tmp = th;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.45d-13) then
        tmp = th
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e-13) {
		tmp = th;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.45e-13:
		tmp = th
	else:
		tmp = 0.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.45e-13)
		tmp = th;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.45e-13)
		tmp = th;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.45e-13], th, 0.0]

Derivation

1. Split input into 2 regimes

2. if kx < 1.4499999999999999e-13

   1. Initial program 88.8%

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

      1. associate-/r/ 88.7%

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

      2. +-commutative 88.7%

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

      3. unpow2 88.7%

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

      4. unpow2 88.7%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 30.1%

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

   5. Taylor expanded in th around 0 17.3%

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

   if 1.4499999999999999e-13 < kx

   1. Initial program 99.4%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 7.2%

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

      1. associate-/r/ 7.2%

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

      2. *-inverses 7.2%

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

      3. *-un-lft-identity 7.2%

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

      4. add-log-exp 19.5%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   6. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   7. Taylor expanded in th around 0 17.8%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 13.7% 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 91.6%

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

   1. associate-/r/ 91.5%

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

   2. +-commutative 91.5%

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

   3. unpow2 91.5%

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

   4. unpow2 91.5%

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

   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 kx around 0 24.2%

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

5. Taylor expanded in th around 0 14.2%

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

6. Final simplification 14.2%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023213 
(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)))