Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 23.9s

Alternatives: 20

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. +-commutative 92.9%

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

   2. unpow2 92.9%

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

   3. unpow2 92.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ t_2 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.77:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.78:\\ \;\;\;\;\sin ky \cdot t_2\\ \mathbf{elif}\;\sin kx \leq 0.9:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\frac{1}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th)))
        (t_2 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.77)
     t_1
     (if (<= (sin kx) -0.02)
       (* ky (/ (sin th) (fabs (sin kx))))
       (if (<= (sin kx) 4e-7)
         (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
         (if (<= (sin kx) 0.78)
           (* (sin ky) t_2)
           (if (<= (sin kx) 0.9) t_1 (/ t_2 (/ 1.0 (sin ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	double t_2 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.77) {
		tmp = t_1;
	} else if (sin(kx) <= -0.02) {
		tmp = ky * (sin(th) / fabs(sin(kx)));
	} else if (sin(kx) <= 4e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (sin(kx) <= 0.78) {
		tmp = sin(ky) * t_2;
	} else if (sin(kx) <= 0.9) {
		tmp = t_1;
	} else {
		tmp = t_2 / (1.0 / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	double t_2 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.77) {
		tmp = t_1;
	} else if (Math.sin(kx) <= -0.02) {
		tmp = ky * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(kx) <= 0.78) {
		tmp = Math.sin(ky) * t_2;
	} else if (Math.sin(kx) <= 0.9) {
		tmp = t_1;
	} else {
		tmp = t_2 / (1.0 / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	t_2 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.77:
		tmp = t_1
	elif math.sin(kx) <= -0.02:
		tmp = ky * (math.sin(th) / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 4e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif math.sin(kx) <= 0.78:
		tmp = math.sin(ky) * t_2
	elif math.sin(kx) <= 0.9:
		tmp = t_1
	else:
		tmp = t_2 / (1.0 / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th))
	t_2 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.77)
		tmp = t_1;
	elseif (sin(kx) <= -0.02)
		tmp = Float64(ky * Float64(sin(th) / abs(sin(kx))));
	elseif (sin(kx) <= 4e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (sin(kx) <= 0.78)
		tmp = Float64(sin(ky) * t_2);
	elseif (sin(kx) <= 0.9)
		tmp = t_1;
	else
		tmp = Float64(t_2 / Float64(1.0 / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	t_2 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.77)
		tmp = t_1;
	elseif (sin(kx) <= -0.02)
		tmp = ky * (sin(th) / abs(sin(kx)));
	elseif (sin(kx) <= 4e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (sin(kx) <= 0.78)
		tmp = sin(ky) * t_2;
	elseif (sin(kx) <= 0.9)
		tmp = t_1;
	else
		tmp = t_2 / (1.0 / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.77], t$95$1, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.78], N[(N[Sin[ky], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.9], t$95$1, N[(t$95$2 / N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 kx) < -0.77000000000000002 or 0.78000000000000003 < (sin.f64 kx) < 0.900000000000000022

   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 99.6%

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

   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 th around 0 54.2%

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

      1. associate-*r/ 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

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

      4. hypot-def 54.3%

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

      5. *-rgt-identity 54.3%

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

      6. hypot-def 54.2%

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

      7. unpow2 54.2%

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

      8. unpow2 54.2%

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

      9. +-commutative 54.2%

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

      10. unpow2 54.2%

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

      11. unpow2 54.2%

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

      12. hypot-def 54.3%

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

   6. Simplified 54.3%

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

   if -0.77000000000000002 < (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 8.3%

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

      1. associate-/l* 8.3%

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

      2. associate-/r/ 8.3%

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

   8. Simplified 8.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 49.6%

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

      3. rem-sqrt-square 49.6%

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

   10. Applied egg-rr 49.6%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 3.9999999999999998e-7

   1. Initial program 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. unpow2 86.0%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 99.9%

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

   if 3.9999999999999998e-7 < (sin.f64 kx) < 0.78000000000000003

   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/ 99.5%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 83.4%

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

   if 0.900000000000000022 < (sin.f64 kx)

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      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 99.4%

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

      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. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-/r/ 99.7%

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

      5. div-inv 99.6%

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

      6. associate-/r* 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0 60.6%

      \[\leadsto \frac{\color{blue}{\frac{\sin th}{\sin kx}}}{\frac{1}{\sin ky}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.77:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.78:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.9:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{1}{\sin ky}}\\ \end{array} \]

Alternative 3: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.77:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.78:\\ \;\;\;\;\sin ky \cdot t_1\\ \mathbf{elif}\;\sin kx \leq 0.9:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{1}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.77)
     (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))
     (if (<= (sin kx) -0.02)
       (* ky (/ (sin th) (fabs (sin kx))))
       (if (<= (sin kx) 4e-7)
         (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
         (if (<= (sin kx) 0.78)
           (* (sin ky) t_1)
           (if (<= (sin kx) 0.9)
             (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
             (/ t_1 (/ 1.0 (sin ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.77) {
		tmp = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	} else if (sin(kx) <= -0.02) {
		tmp = ky * (sin(th) / fabs(sin(kx)));
	} else if (sin(kx) <= 4e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (sin(kx) <= 0.78) {
		tmp = sin(ky) * t_1;
	} else if (sin(kx) <= 0.9) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = t_1 / (1.0 / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.77) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(kx) <= -0.02) {
		tmp = ky * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(kx) <= 0.78) {
		tmp = Math.sin(ky) * t_1;
	} else if (Math.sin(kx) <= 0.9) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = t_1 / (1.0 / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.77:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(kx) <= -0.02:
		tmp = ky * (math.sin(th) / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 4e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif math.sin(kx) <= 0.78:
		tmp = math.sin(ky) * t_1
	elif math.sin(kx) <= 0.9:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = t_1 / (1.0 / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.77)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)));
	elseif (sin(kx) <= -0.02)
		tmp = Float64(ky * Float64(sin(th) / abs(sin(kx))));
	elseif (sin(kx) <= 4e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (sin(kx) <= 0.78)
		tmp = Float64(sin(ky) * t_1);
	elseif (sin(kx) <= 0.9)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(t_1 / Float64(1.0 / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.77)
		tmp = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	elseif (sin(kx) <= -0.02)
		tmp = ky * (sin(th) / abs(sin(kx)));
	elseif (sin(kx) <= 4e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (sin(kx) <= 0.78)
		tmp = sin(ky) * t_1;
	elseif (sin(kx) <= 0.9)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = t_1 / (1.0 / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.77], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.78], N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.9], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (sin.f64 kx) < -0.77000000000000002

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.6%

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

   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 th around 0 53.5%

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

   if -0.77000000000000002 < (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 8.3%

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

      1. associate-/l* 8.3%

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

      2. associate-/r/ 8.3%

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

   8. Simplified 8.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 49.6%

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

      3. rem-sqrt-square 49.6%

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

   10. Applied egg-rr 49.6%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 3.9999999999999998e-7

   1. Initial program 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. unpow2 86.0%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 99.9%

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

   if 3.9999999999999998e-7 < (sin.f64 kx) < 0.78000000000000003

   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/ 99.5%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 83.4%

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

   if 0.78000000000000003 < (sin.f64 kx) < 0.900000000000000022

   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/ 99.6%

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

      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 99.8%

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

   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 th around 0 58.0%

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

      1. associate-*r/ 58.0%

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

      2. unpow2 58.0%

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

      3. unpow2 58.0%

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

      4. hypot-def 58.3%

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

      5. *-rgt-identity 58.3%

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

      6. hypot-def 58.0%

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

      7. unpow2 58.0%

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

      8. unpow2 58.0%

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

      9. +-commutative 58.0%

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

      10. unpow2 58.0%

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

      11. unpow2 58.0%

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

      12. hypot-def 58.3%

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

   6. Simplified 58.3%

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

   if 0.900000000000000022 < (sin.f64 kx)

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      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 99.4%

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

      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. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-/r/ 99.7%

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

      5. div-inv 99.6%

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

      6. associate-/r* 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0 60.6%

      \[\leadsto \frac{\color{blue}{\frac{\sin th}{\sin kx}}}{\frac{1}{\sin ky}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.77:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.78:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.9:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{1}{\sin ky}}\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.68:\\ \;\;\;\;\frac{\frac{\sin ky}{t_1}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \left|\frac{1}{\sin ky}\right|\right)\\ \mathbf{elif}\;\sin ky \leq 0.0001:\\ \;\;\;\;\frac{\frac{\sin th}{t_1}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.68)
     (/ (/ (sin ky) t_1) (+ (/ 1.0 th) (* th 0.16666666666666666)))
     (if (<= (sin ky) -0.0005)
       (* (sin th) (* (sin ky) (fabs (/ 1.0 (sin ky)))))
       (if (<= (sin ky) 0.0001)
         (/ (/ (sin th) t_1) (+ (* ky 0.16666666666666666) (/ 1.0 ky)))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.68) {
		tmp = (sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	} else if (sin(ky) <= -0.0005) {
		tmp = sin(th) * (sin(ky) * fabs((1.0 / sin(ky))));
	} else if (sin(ky) <= 0.0001) {
		tmp = (sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -0.68) {
		tmp = (Math.sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	} else if (Math.sin(ky) <= -0.0005) {
		tmp = Math.sin(th) * (Math.sin(ky) * Math.abs((1.0 / Math.sin(ky))));
	} else if (Math.sin(ky) <= 0.0001) {
		tmp = (Math.sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.68:
		tmp = (math.sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666))
	elif math.sin(ky) <= -0.0005:
		tmp = math.sin(th) * (math.sin(ky) * math.fabs((1.0 / math.sin(ky))))
	elif math.sin(ky) <= 0.0001:
		tmp = (math.sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.68)
		tmp = Float64(Float64(sin(ky) / t_1) / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)));
	elseif (sin(ky) <= -0.0005)
		tmp = Float64(sin(th) * Float64(sin(ky) * abs(Float64(1.0 / sin(ky)))));
	elseif (sin(ky) <= 0.0001)
		tmp = Float64(Float64(sin(th) / t_1) / Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.68)
		tmp = (sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	elseif (sin(ky) <= -0.0005)
		tmp = sin(th) * (sin(ky) * abs((1.0 / sin(ky))));
	elseif (sin(ky) <= 0.0001)
		tmp = (sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.68], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0005], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0001], N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.680000000000000049

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r/ 99.4%

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

      2. div-inv 99.3%

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

      3. associate-/r* 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 55.4%

      \[\leadsto \frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\color{blue}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   7. Step-by-step derivation

      1. *-commutative 55.4%

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

   8. Simplified 55.4%

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

   if -0.680000000000000049 < (sin.f64 ky) < -5.0000000000000001e-4

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. associate-*r/ 2.8%

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

      2. *-commutative 2.8%

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

      3. div-inv 2.8%

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

      4. associate-*l* 2.8%

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

   6. Applied egg-rr 2.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 72.5%

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

      3. inv-pow 72.5%

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

      4. inv-pow 72.5%

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

      5. pow-prod-up 72.5%

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

      6. metadata-eval 72.5%

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

   8. Applied egg-rr 72.5%

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

      1. metadata-eval 72.5%

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

      2. pow-sqr 72.5%

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

      3. unpow-1 72.5%

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

      4. unpow-1 72.5%

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

      5. rem-sqrt-square 72.5%

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

   10. Simplified 72.5%

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

   if -5.0000000000000001e-4 < (sin.f64 ky) < 1.00000000000000005e-4

   1. Initial program 86.6%

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

      1. associate-*l/ 83.7%

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

      2. associate-*r/ 86.6%

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

      3. +-commutative 86.6%

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

      4. unpow2 86.6%

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

      5. unpow2 86.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. associate-/r/ 99.6%

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

      5. div-inv 99.4%

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

      6. associate-/r* 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in ky around 0 99.5%

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

   if 1.00000000000000005e-4 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 69.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.68:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \left|\frac{1}{\sin ky}\right|\right)\\ \mathbf{elif}\;\sin ky \leq 0.0001:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 40.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-175}:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin ky) -1e-5)
     t_1
     (if (<= (sin ky) -2e-153)
       (/ (* ky (sin th)) kx)
       (if (<= (sin ky) -1e-175)
         (- th)
         (if (<= (sin ky) -2e-205)
           t_1
           (if (<= (sin ky) 5e-97) (/ (sin th) (/ kx ky)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(ky) <= -1e-5) {
		tmp = t_1;
	} else if (sin(ky) <= -2e-153) {
		tmp = (ky * sin(th)) / kx;
	} else if (sin(ky) <= -1e-175) {
		tmp = -th;
	} else if (sin(ky) <= -2e-205) {
		tmp = t_1;
	} else if (sin(ky) <= 5e-97) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (sin(ky) <= (-1d-5)) then
        tmp = t_1
    else if (sin(ky) <= (-2d-153)) then
        tmp = (ky * sin(th)) / kx
    else if (sin(ky) <= (-1d-175)) then
        tmp = -th
    else if (sin(ky) <= (-2d-205)) then
        tmp = t_1
    else if (sin(ky) <= 5d-97) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (Math.sin(ky) <= -1e-5) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -2e-153) {
		tmp = (ky * Math.sin(th)) / kx;
	} else if (Math.sin(ky) <= -1e-175) {
		tmp = -th;
	} else if (Math.sin(ky) <= -2e-205) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 5e-97) {
		tmp = Math.sin(th) / (kx / ky);
	} 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) <= -1e-5:
		tmp = t_1
	elif math.sin(ky) <= -2e-153:
		tmp = (ky * math.sin(th)) / kx
	elif math.sin(ky) <= -1e-175:
		tmp = -th
	elif math.sin(ky) <= -2e-205:
		tmp = t_1
	elif math.sin(ky) <= 5e-97:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (sin(ky) <= -1e-5)
		tmp = t_1;
	elseif (sin(ky) <= -2e-153)
		tmp = Float64(Float64(ky * sin(th)) / kx);
	elseif (sin(ky) <= -1e-175)
		tmp = Float64(-th);
	elseif (sin(ky) <= -2e-205)
		tmp = t_1;
	elseif (sin(ky) <= 5e-97)
		tmp = Float64(sin(th) / Float64(kx / ky));
	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) <= -1e-5)
		tmp = t_1;
	elseif (sin(ky) <= -2e-153)
		tmp = (ky * sin(th)) / kx;
	elseif (sin(ky) <= -1e-175)
		tmp = -th;
	elseif (sin(ky) <= -2e-205)
		tmp = t_1;
	elseif (sin(ky) <= 5e-97)
		tmp = sin(th) / (kx / ky);
	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], -1e-5], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-153], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-175], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-205], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-97], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -1.00000000000000008e-5 or -1e-175 < (sin.f64 ky) < -2e-205

   1. Initial program 96.9%

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

      1. associate-*l/ 96.7%

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

      2. associate-*r/ 96.7%

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

      3. +-commutative 96.7%

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

      4. unpow2 96.7%

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

      5. unpow2 96.7%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 25.7%

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

      3. pow2 25.7%

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

   6. Applied egg-rr 25.7%

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

      1. *-commutative 25.7%

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

      2. associate-/r/ 25.8%

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

      3. *-inverses 25.8%

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

      4. /-rgt-identity 25.8%

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

      5. unpow2 25.8%

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

      6. rem-sqrt-square 31.6%

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

   8. Simplified 31.6%

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

   if -1.00000000000000008e-5 < (sin.f64 ky) < -2.00000000000000008e-153

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in ky around 0 42.4%

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

      1. associate-/l* 42.2%

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

      2. associate-/r/ 42.2%

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

   8. Simplified 42.2%

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

   9. Taylor expanded in kx around 0 12.6%

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

   if -2.00000000000000008e-153 < (sin.f64 ky) < -1e-175

   1. Initial program 2.4%

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

      1. associate-*l/ 2.0%

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

      2. +-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. unpow2 2.0%

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

      5. hypot-def 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in kx around 0 3.1%

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

   5. Taylor expanded in th around 0 3.1%

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

      1. associate-/l* 2.7%

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

      2. associate-/r/ 2.7%

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

      3. *-un-lft-identity 2.7%

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

      4. associate-*l/ 2.7%

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

      5. lft-mult-inverse 2.7%

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

      6. *-un-lft-identity 2.7%

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

      7. add-sqr-sqrt 2.2%

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

      8. sqrt-unprod 27.6%

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

   7. Applied egg-rr 27.6%

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

   8. Taylor expanded in th around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -2e-205 < (sin.f64 ky) < 4.9999999999999995e-97

   1. Initial program 82.1%

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

      1. +-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. unpow2 82.1%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in ky around 0 62.4%

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

      1. associate-/l* 64.8%

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

      2. associate-/r/ 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in kx around 0 45.6%

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

       1. associate-/l* 48.0%

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

   11. Simplified 48.0%

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

   if 4.9999999999999995e-97 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 62.0%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-175}:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (* ky (/ (sin th) (fabs (sin kx))))
   (if (<= (sin kx) 4e-7)
     (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = ky * (sin(th) / fabs(sin(kx)));
	} else if (sin(kx) <= 4e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = ky * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} 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) <= -0.02:
		tmp = ky * (math.sin(th) / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 4e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	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) <= -0.02)
		tmp = Float64(ky * Float64(sin(th) / abs(sin(kx))));
	elseif (sin(kx) <= 4e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	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) <= -0.02)
		tmp = ky * (sin(th) / abs(sin(kx)));
	elseif (sin(kx) <= 4e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $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) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 12.1%

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

      1. associate-/l* 12.1%

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

      2. associate-/r/ 12.1%

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

   8. Simplified 12.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 47.9%

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

      3. rem-sqrt-square 47.9%

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

   10. Applied egg-rr 47.9%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 3.9999999999999998e-7

   1. Initial program 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. unpow2 86.0%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 99.9%

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

   if 3.9999999999999998e-7 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 71.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 7: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0006:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-14}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.0006)
   (fabs (sin th))
   (if (<= (sin ky) 1e-14) (* ky (/ (sin th) (fabs (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.0006) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-14) {
		tmp = 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) :: tmp
    if (sin(ky) <= (-0.0006d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-14) then
        tmp = 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 tmp;
	if (Math.sin(ky) <= -0.0006) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-14) {
		tmp = ky * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.0006:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-14:
		tmp = ky * (math.sin(th) / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.0006)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-14)
		tmp = Float64(ky * Float64(sin(th) / abs(sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.0006)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-14)
		tmp = ky * (sin(th) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0006], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-14], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.99999999999999947e-4

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 2.7%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 25.4%

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

      3. pow2 25.4%

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

   6. Applied egg-rr 25.4%

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

      1. *-commutative 25.4%

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

      2. associate-/r/ 25.4%

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

      3. *-inverses 25.4%

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

      4. /-rgt-identity 25.4%

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

      5. unpow2 25.4%

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

      6. rem-sqrt-square 32.0%

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

   8. Simplified 32.0%

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

   if -5.99999999999999947e-4 < (sin.f64 ky) < 9.99999999999999999e-15

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

      2. unpow2 86.3%

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

      3. unpow2 86.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 48.3%

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

      1. associate-/l* 49.5%

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

      2. associate-/r/ 49.4%

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

   8. Simplified 49.4%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-prod 73.5%

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

      3. rem-sqrt-square 79.3%

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

   10. Applied egg-rr 79.3%

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

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

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 69.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0006:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-14}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. associate-*l/ 91.3%

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

   2. associate-*r/ 92.9%

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

   3. +-commutative 92.9%

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

   4. unpow2 92.9%

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

   5. unpow2 92.9%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 9: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -350:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;th \leq 0.054:\\ \;\;\;\;\frac{\frac{\sin ky}{t_1}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= th -350.0)
     (/ (* ky (sin th)) t_1)
     (if (<= th 0.054)
       (/ (/ (sin ky) t_1) (+ (/ 1.0 th) (* th 0.16666666666666666)))
       (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (th <= -350.0) {
		tmp = (ky * sin(th)) / t_1;
	} else if (th <= 0.054) {
		tmp = (sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), 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 (th <= -350.0) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else if (th <= 0.054) {
		tmp = (Math.sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if th <= -350.0:
		tmp = (ky * math.sin(th)) / t_1
	elif th <= 0.054:
		tmp = (math.sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (th <= -350.0)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	elseif (th <= 0.054)
		tmp = Float64(Float64(sin(ky) / t_1) / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (th <= -350.0)
		tmp = (ky * sin(th)) / t_1;
	elseif (th <= 0.054)
		tmp = (sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	else
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), 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[th, -350.0], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[th, 0.054], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < -350

   1. Initial program 92.0%

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

      1. associate-*l/ 92.0%

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

      2. +-commutative 92.0%

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

      3. unpow2 92.0%

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

      4. unpow2 92.0%

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

      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 64.5%

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

   if -350 < th < 0.0539999999999999994

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. unpow2 91.3%

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

      3. unpow2 91.3%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r/ 99.7%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around 0 98.6%

      \[\leadsto \frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\color{blue}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   7. Step-by-step derivation

      1. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   if 0.0539999999999999994 < th

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

      3. unpow2 97.9%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 57.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -350:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 0.054:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]

Alternative 10: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq -350:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 0.00033:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th -350.0)
   (/ (* ky (sin th)) (hypot (sin ky) (sin kx)))
   (if (<= th 0.00033)
     (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
     (* (sin th) (/ (sin ky) (hypot (sin ky) kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= -350.0) {
		tmp = (ky * sin(th)) / hypot(sin(ky), sin(kx));
	} else if (th <= 0.00033) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= -350.0) {
		tmp = (ky * Math.sin(th)) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (th <= 0.00033) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= -350.0:
		tmp = (ky * math.sin(th)) / math.hypot(math.sin(ky), math.sin(kx))
	elif th <= 0.00033:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= -350.0)
		tmp = Float64(Float64(ky * sin(th)) / hypot(sin(ky), sin(kx)));
	elseif (th <= 0.00033)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= -350.0)
		tmp = (ky * sin(th)) / hypot(sin(ky), sin(kx));
	elseif (th <= 0.00033)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, -350.0], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 0.00033], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < -350

   1. Initial program 92.0%

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

      1. associate-*l/ 92.0%

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

      2. +-commutative 92.0%

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

      3. unpow2 92.0%

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

      4. unpow2 92.0%

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

      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 64.5%

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

   if -350 < th < 3.3e-4

   1. Initial program 91.3%

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

      1. associate-/r/ 91.2%

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

      2. +-commutative 91.2%

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

      3. unpow2 91.2%

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

      4. unpow2 91.2%

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

      5. hypot-def 99.7%

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

   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 th around 0 89.7%

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

      1. associate-*r/ 89.8%

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

      2. unpow2 89.8%

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

      3. unpow2 89.8%

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

      4. hypot-def 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. hypot-def 89.8%

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

      7. unpow2 89.8%

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

      8. unpow2 89.8%

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

      9. +-commutative 89.8%

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

      10. unpow2 89.8%

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

      11. unpow2 89.8%

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

      12. hypot-def 98.3%

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

   6. Simplified 98.3%

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

   if 3.3e-4 < th

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

      3. unpow2 97.9%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 57.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -350:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 0.00033:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]

Alternative 11: 48.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-31)
   (fabs (sin th))
   (if (<= (sin ky) 5e-97) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-31) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-97) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-5d-31)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-97) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -5e-31) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-97) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-31:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-97:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-97)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-97)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-31], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-97], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5e-31

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 2.7%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 23.9%

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

      3. pow2 23.9%

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

   6. Applied egg-rr 23.9%

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

      1. *-commutative 23.9%

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

      2. associate-/r/ 24.0%

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

      3. *-inverses 24.0%

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

      4. /-rgt-identity 24.0%

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

      5. unpow2 24.0%

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

      6. rem-sqrt-square 30.3%

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

   8. Simplified 30.3%

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

   if -5e-31 < (sin.f64 ky) < 4.9999999999999995e-97

   1. Initial program 82.9%

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

      1. +-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 53.9%

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

   if 4.9999999999999995e-97 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 62.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 48.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-31)
   (fabs (sin th))
   (if (<= (sin ky) 5e-97) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-31) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-97) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-5d-31)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-97) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -5e-31) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-97) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-31:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-97:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-97)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-97)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-31], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-97], N[(ky * 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) < -5e-31

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 2.7%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 23.9%

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

      3. pow2 23.9%

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

   6. Applied egg-rr 23.9%

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

      1. *-commutative 23.9%

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

      2. associate-/r/ 24.0%

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

      3. *-inverses 24.0%

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

      4. /-rgt-identity 24.0%

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

      5. unpow2 24.0%

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

      6. rem-sqrt-square 30.3%

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

   8. Simplified 30.3%

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

   if -5e-31 < (sin.f64 ky) < 4.9999999999999995e-97

   1. Initial program 82.9%

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

      1. +-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 52.4%

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

      1. associate-/l* 53.9%

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

      2. associate-/r/ 53.8%

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

   8. Simplified 53.8%

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

   if 4.9999999999999995e-97 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 62.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 48.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-31)
   (fabs (sin th))
   (if (<= (sin ky) 5e-97) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-31) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-97) {
		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) <= (-5d-31)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-97) 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) <= -5e-31) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-97) {
		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) <= -5e-31:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-97:
		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) <= -5e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-97)
		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) <= -5e-31)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-97)
		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], -5e-31], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-97], 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) < -5e-31

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 2.7%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 23.9%

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

      3. pow2 23.9%

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

   6. Applied egg-rr 23.9%

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

      1. *-commutative 23.9%

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

      2. associate-/r/ 24.0%

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

      3. *-inverses 24.0%

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

      4. /-rgt-identity 24.0%

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

      5. unpow2 24.0%

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

      6. rem-sqrt-square 30.3%

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

   8. Simplified 30.3%

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

   if -5e-31 < (sin.f64 ky) < 4.9999999999999995e-97

   1. Initial program 82.9%

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

      1. +-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 52.4%

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

      1. associate-/l* 53.9%

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

   6. Simplified 53.9%

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

   if 4.9999999999999995e-97 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 62.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 32.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \frac{th}{\sin kx}\\ \mathbf{if}\;ky \leq -580000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.9 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -1.45 \cdot 10^{-214}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 1.5 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (/ th (sin kx)))))
   (if (<= ky -580000000.0)
     (sin th)
     (if (<= ky -3.9e-163)
       t_1
       (if (<= ky -1.45e-214) (- th) (if (<= ky 1.5e-96) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * (th / sin(kx));
	double tmp;
	if (ky <= -580000000.0) {
		tmp = sin(th);
	} else if (ky <= -3.9e-163) {
		tmp = t_1;
	} else if (ky <= -1.45e-214) {
		tmp = -th;
	} else if (ky <= 1.5e-96) {
		tmp = t_1;
	} 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 = ky * (th / sin(kx))
    if (ky <= (-580000000.0d0)) then
        tmp = sin(th)
    else if (ky <= (-3.9d-163)) then
        tmp = t_1
    else if (ky <= (-1.45d-214)) then
        tmp = -th
    else if (ky <= 1.5d-96) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * (th / Math.sin(kx));
	double tmp;
	if (ky <= -580000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= -3.9e-163) {
		tmp = t_1;
	} else if (ky <= -1.45e-214) {
		tmp = -th;
	} else if (ky <= 1.5e-96) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * (th / math.sin(kx))
	tmp = 0
	if ky <= -580000000.0:
		tmp = math.sin(th)
	elif ky <= -3.9e-163:
		tmp = t_1
	elif ky <= -1.45e-214:
		tmp = -th
	elif ky <= 1.5e-96:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * Float64(th / sin(kx)))
	tmp = 0.0
	if (ky <= -580000000.0)
		tmp = sin(th);
	elseif (ky <= -3.9e-163)
		tmp = t_1;
	elseif (ky <= -1.45e-214)
		tmp = Float64(-th);
	elseif (ky <= 1.5e-96)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * (th / sin(kx));
	tmp = 0.0;
	if (ky <= -580000000.0)
		tmp = sin(th);
	elseif (ky <= -3.9e-163)
		tmp = t_1;
	elseif (ky <= -1.45e-214)
		tmp = -th;
	elseif (ky <= 1.5e-96)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ky, -580000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -3.9e-163], t$95$1, If[LessEqual[ky, -1.45e-214], (-th), If[LessEqual[ky, 1.5e-96], t$95$1, N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -5.8e8 or 1.5e-96 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 38.0%

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

   if -5.8e8 < ky < -3.9000000000000002e-163 or -1.44999999999999993e-214 < ky < 1.5e-96

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. unpow2 89.0%

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

      3. unpow2 89.0%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 53.8%

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

      1. associate-/l* 55.3%

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

      2. associate-/r/ 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in th around 0 31.0%

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

   if -3.9000000000000002e-163 < ky < -1.44999999999999993e-214

   1. Initial program 42.8%

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

      1. associate-*l/ 42.6%

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

      2. +-commutative 42.6%

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

      3. unpow2 42.6%

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

      4. unpow2 42.6%

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

      5. hypot-def 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in kx around 0 2.6%

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

   5. Taylor expanded in th around 0 2.6%

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

      1. associate-/l* 2.2%

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

      2. associate-/r/ 2.4%

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

      3. *-un-lft-identity 2.4%

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

      4. associate-*l/ 2.4%

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

      5. lft-mult-inverse 2.4%

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

      6. *-un-lft-identity 2.4%

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

      7. add-sqr-sqrt 1.2%

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

      8. sqrt-unprod 10.9%

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

   7. Applied egg-rr 10.9%

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

   8. Taylor expanded in th around -inf 43.0%

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

      1. neg-mul-1 43.0%

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

   10. Simplified 43.0%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -580000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.9 \cdot 10^{-163}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;ky \leq -1.45 \cdot 10^{-214}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 1.5 \cdot 10^{-96}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 34.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -840000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -840000000.0)
   (sin th)
   (if (<= ky 1.05e-96) (* ky (/ (sin th) kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -840000000.0) {
		tmp = sin(th);
	} else if (ky <= 1.05e-96) {
		tmp = ky * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-840000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 1.05d-96) then
        tmp = ky * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -840000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1.05e-96) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -840000000.0:
		tmp = math.sin(th)
	elif ky <= 1.05e-96:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -840000000.0)
		tmp = sin(th);
	elseif (ky <= 1.05e-96)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -840000000.0)
		tmp = sin(th);
	elseif (ky <= 1.05e-96)
		tmp = ky * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -840000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.05e-96], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -8.4e8 or 1.05000000000000001e-96 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 38.0%

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

   if -8.4e8 < ky < 1.05000000000000001e-96

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. unpow2 83.9%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 49.3%

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

      1. associate-/l* 50.6%

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

      2. associate-/r/ 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in kx around 0 32.4%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -840000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 34.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -840000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -840000000.0)
   (sin th)
   (if (<= ky 8e-96) (/ (sin th) (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -840000000.0) {
		tmp = sin(th);
	} else if (ky <= 8e-96) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-840000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 8d-96) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -840000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 8e-96) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -840000000.0:
		tmp = math.sin(th)
	elif ky <= 8e-96:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -840000000.0)
		tmp = sin(th);
	elseif (ky <= 8e-96)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -840000000.0)
		tmp = sin(th);
	elseif (ky <= 8e-96)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -840000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 8e-96], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -8.4e8 or 7.9999999999999993e-96 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 38.0%

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

   if -8.4e8 < ky < 7.9999999999999993e-96

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. unpow2 83.9%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 49.3%

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

      1. associate-/l* 50.6%

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

      2. associate-/r/ 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in kx around 0 31.1%

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

       1. associate-/l* 32.4%

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

   11. Simplified 32.4%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -840000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 26.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -840000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -4.5 \cdot 10^{-296}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -840000000.0) (sin th) (if (<= ky -4.5e-296) (- th) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -840000000.0) {
		tmp = sin(th);
	} else if (ky <= -4.5e-296) {
		tmp = -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 <= (-840000000.0d0)) then
        tmp = sin(th)
    else if (ky <= (-4.5d-296)) then
        tmp = -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 <= -840000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= -4.5e-296) {
		tmp = -th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -840000000.0:
		tmp = math.sin(th)
	elif ky <= -4.5e-296:
		tmp = -th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -840000000.0)
		tmp = sin(th);
	elseif (ky <= -4.5e-296)
		tmp = Float64(-th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -840000000.0)
		tmp = sin(th);
	elseif (ky <= -4.5e-296)
		tmp = -th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -840000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -4.5e-296], (-th), N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -8.4e8 or -4.5000000000000002e-296 < ky

   1. Initial program 96.8%

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

      1. +-commutative 96.8%

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

      2. unpow2 96.8%

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

      3. unpow2 96.8%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 32.9%

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

   if -8.4e8 < ky < -4.5000000000000002e-296

   1. Initial program 82.1%

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

      1. associate-*l/ 78.7%

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

      2. +-commutative 78.7%

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

      3. unpow2 78.7%

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

      4. unpow2 78.7%

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

      5. hypot-def 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in kx around 0 11.4%

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

   5. Taylor expanded in th around 0 11.8%

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

      1. associate-/l* 3.2%

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

      2. associate-/r/ 3.3%

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

      3. *-un-lft-identity 3.3%

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

      4. associate-*l/ 3.3%

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

      5. lft-mult-inverse 3.3%

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

      6. *-un-lft-identity 3.3%

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

      7. add-sqr-sqrt 1.2%

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

      8. sqrt-unprod 11.6%

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

   7. Applied egg-rr 11.6%

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

   8. Taylor expanded in th around -inf 15.9%

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

      1. neg-mul-1 15.9%

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

   10. Simplified 15.9%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -840000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -4.5 \cdot 10^{-296}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18: 25.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.5e-6) (sin th) (sqrt (* th th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.5e-6) {
		tmp = sin(th);
	} else {
		tmp = sqrt((th * th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 3.5d-6) then
        tmp = sin(th)
    else
        tmp = sqrt((th * th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.5e-6) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sqrt((th * th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.5e-6:
		tmp = math.sin(th)
	else:
		tmp = math.sqrt((th * th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.5e-6)
		tmp = sin(th);
	else
		tmp = sqrt(Float64(th * th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.5e-6)
		tmp = sin(th);
	else
		tmp = sqrt((th * th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.5e-6], N[Sin[th], $MachinePrecision], N[Sqrt[N[(th * th), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.49999999999999995e-6

   1. Initial program 90.9%

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

      1. +-commutative 90.9%

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

      2. unpow2 90.9%

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

      3. unpow2 90.9%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 31.1%

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

   if 3.49999999999999995e-6 < 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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 15.9%

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

   5. Taylor expanded in th around 0 15.2%

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

      1. associate-/l* 4.3%

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

      2. associate-/r/ 4.4%

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

      3. *-un-lft-identity 4.4%

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

      4. associate-*l/ 4.4%

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

      5. lft-mult-inverse 4.4%

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

      6. *-un-lft-identity 4.4%

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

      7. add-sqr-sqrt 1.8%

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

      8. sqrt-unprod 14.7%

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

   7. Applied egg-rr 14.7%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \end{array} \]

Alternative 19: 17.1% accurate, 174.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.55 \cdot 10^{-298}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= ky -1.55e-298) (- th) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.55e-298) {
		tmp = -th;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.55d-298)) then
        tmp = -th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.55e-298) {
		tmp = -th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.55e-298:
		tmp = -th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.55e-298)
		tmp = Float64(-th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.55e-298)
		tmp = -th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.55e-298], (-th), th]

Derivation

1. Split input into 2 regimes

2. if ky < -1.5500000000000001e-298

   1. Initial program 89.9%

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

      1. associate-*l/ 88.0%

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

      2. +-commutative 88.0%

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

      3. unpow2 88.0%

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

      4. unpow2 88.0%

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

      5. hypot-def 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in kx around 0 20.4%

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

   5. Taylor expanded in th around 0 11.7%

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

      1. associate-/l* 6.9%

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

      2. associate-/r/ 6.9%

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

      3. *-un-lft-identity 6.9%

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

      4. associate-*l/ 6.9%

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

      5. lft-mult-inverse 6.9%

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

      6. *-un-lft-identity 6.9%

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

      7. add-sqr-sqrt 3.5%

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

      8. sqrt-unprod 10.4%

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

   7. Applied egg-rr 10.4%

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

   8. Taylor expanded in th around -inf 17.1%

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

      1. neg-mul-1 17.1%

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

   10. Simplified 17.1%

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

   if -1.5500000000000001e-298 < ky

   1. Initial program 95.6%

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

      1. associate-*l/ 94.3%

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

      2. associate-*r/ 95.6%

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

      3. +-commutative 95.6%

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

      4. unpow2 95.6%

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

      5. unpow2 95.6%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 33.3%

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

   5. Taylor expanded in th around 0 15.7%

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

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.55 \cdot 10^{-298}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 20: 13.6% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.9%

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

   1. associate-*l/ 91.3%

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

   2. associate-*r/ 92.9%

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

   3. +-commutative 92.9%

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

   4. unpow2 92.9%

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

   5. unpow2 92.9%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in kx around 0 25.0%

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

5. Taylor expanded in th around 0 11.6%

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

6. Final simplification 11.6%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023260 
(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)))