Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 11.3s

Alternatives: 19

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 95.2

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 99.7

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

3. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ t_2 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.4:\\ \;\;\;\;th \cdot \frac{\sin ky}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 10^{-6}:\\ \;\;\;\;\left(\sqrt{\frac{1}{t\_3}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;\frac{th}{\frac{t\_2}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot kx (sin ky))) (sin th)))
        (t_2 (hypot (sin kx) (sin ky)))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) t_3)))))
   (if (<= t_4 -1.0)
     t_1
     (if (<= t_4 -0.4)
       (* th (/ (sin ky) t_2))
       (if (<= t_4 1e-6)
         (* (* (sqrt (/ 1.0 t_3)) (sin ky)) (sin th))
         (if (<= t_4 0.995) (/ th (/ t_2 (sin ky))) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	double t_2 = hypot(sin(kx), sin(ky));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(ky), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_1;
	} else if (t_4 <= -0.4) {
		tmp = th * (sin(ky) / t_2);
	} else if (t_4 <= 1e-6) {
		tmp = (sqrt((1.0 / t_3)) * sin(ky)) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = th / (t_2 / sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(kx, Math.sin(ky))) * Math.sin(th);
	double t_2 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_3 = Math.pow(Math.sin(kx), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_1;
	} else if (t_4 <= -0.4) {
		tmp = th * (Math.sin(ky) / t_2);
	} else if (t_4 <= 1e-6) {
		tmp = (Math.sqrt((1.0 / t_3)) * Math.sin(ky)) * Math.sin(th);
	} else if (t_4 <= 0.995) {
		tmp = th / (t_2 / Math.sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(kx, math.sin(ky))) * math.sin(th)
	t_2 = math.hypot(math.sin(kx), math.sin(ky))
	t_3 = math.pow(math.sin(kx), 2.0)
	t_4 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + t_3))
	tmp = 0
	if t_4 <= -1.0:
		tmp = t_1
	elif t_4 <= -0.4:
		tmp = th * (math.sin(ky) / t_2)
	elif t_4 <= 1e-6:
		tmp = (math.sqrt((1.0 / t_3)) * math.sin(ky)) * math.sin(th)
	elif t_4 <= 0.995:
		tmp = th / (t_2 / math.sin(ky))
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(kx, sin(ky))) * sin(th))
	t_2 = hypot(sin(kx), sin(ky))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = t_1;
	elseif (t_4 <= -0.4)
		tmp = Float64(th * Float64(sin(ky) / t_2));
	elseif (t_4 <= 1e-6)
		tmp = Float64(Float64(sqrt(Float64(1.0 / t_3)) * sin(ky)) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = Float64(th / Float64(t_2 / sin(ky)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	t_2 = hypot(sin(kx), sin(ky));
	t_3 = sin(kx) ^ 2.0;
	t_4 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + t_3));
	tmp = 0.0;
	if (t_4 <= -1.0)
		tmp = t_1;
	elseif (t_4 <= -0.4)
		tmp = th * (sin(ky) / t_2);
	elseif (t_4 <= 1e-6)
		tmp = (sqrt((1.0 / t_3)) * sin(ky)) * sin(th);
	elseif (t_4 <= 0.995)
		tmp = th / (t_2 / sin(ky));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], t$95$1, If[LessEqual[t$95$4, -0.4], N[(th * N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e-6], N[(N[(N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], N[(th / N[(t$95$2 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.3

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 100.0

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 96.2

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.4

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

   3. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 44.8%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 96.4%

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

   if 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-sin-a N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 51.6%

      \[\leadsto \frac{\color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.4:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\left(\sqrt{\frac{1}{{\sin kx}^{2}}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.4:\\ \;\;\;\;th \cdot \frac{\sin ky}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;\frac{th}{\frac{t\_3}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot kx (sin ky))) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.4)
       (* th (/ (sin ky) t_3))
       (if (<= t_2 1e-6)
         (* (/ (sin ky) (hypot (sin kx) ky)) (sin th))
         (if (<= t_2 0.995) (/ th (/ t_3 (sin ky))) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.4) {
		tmp = th * (sin(ky) / t_3);
	} else if (t_2 <= 1e-6) {
		tmp = (sin(ky) / hypot(sin(kx), ky)) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = th / (t_3 / sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(kx, Math.sin(ky))) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_3 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.4) {
		tmp = th * (Math.sin(ky) / t_3);
	} else if (t_2 <= 1e-6) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), ky)) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = th / (t_3 / Math.sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(kx, math.sin(ky))) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_3 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_2 <= -1.0:
		tmp = t_1
	elif t_2 <= -0.4:
		tmp = th * (math.sin(ky) / t_3)
	elif t_2 <= 1e-6:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), ky)) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = th / (t_3 / math.sin(ky))
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(kx, sin(ky))) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.4)
		tmp = Float64(th * Float64(sin(ky) / t_3));
	elseif (t_2 <= 1e-6)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), ky)) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = Float64(th / Float64(t_3 / sin(ky)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_3 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.4)
		tmp = th * (sin(ky) / t_3);
	elseif (t_2 <= 1e-6)
		tmp = (sin(ky) / hypot(sin(kx), ky)) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = th / (t_3 / sin(ky));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.4], N[(th * N[(N[Sin[ky], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], N[(th / N[(t$95$3 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.3

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 100.0

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 96.2

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.4

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

   3. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 44.8%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 95.8%

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

   if 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-sin-a N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 51.6%

      \[\leadsto \frac{\color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.4:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_3 := th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot kx (sin ky))) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_3 (* th (/ (sin ky) (hypot (sin kx) (sin ky))))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.4)
       t_3
       (if (<= t_2 1e-6)
         (* (/ (sin ky) (hypot (sin kx) ky)) (sin th))
         (if (<= t_2 0.995) t_3 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_3 = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.4) {
		tmp = t_3;
	} else if (t_2 <= 1e-6) {
		tmp = (sin(ky) / hypot(sin(kx), ky)) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(kx, Math.sin(ky))) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_3 = th * (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.4) {
		tmp = t_3;
	} else if (t_2 <= 1e-6) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), ky)) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(kx, math.sin(ky))) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_3 = th * (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky)))
	tmp = 0
	if t_2 <= -1.0:
		tmp = t_1
	elif t_2 <= -0.4:
		tmp = t_3
	elif t_2 <= 1e-6:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), ky)) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(kx, sin(ky))) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_3 = Float64(th * Float64(sin(ky) / hypot(sin(kx), sin(ky))))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.4)
		tmp = t_3;
	elseif (t_2 <= 1e-6)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), ky)) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_3 = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.4)
		tmp = t_3;
	elseif (t_2 <= 1e-6)
		tmp = (sin(ky) / hypot(sin(kx), ky)) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.4], t$95$3, If[LessEqual[t$95$2, 1e-6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.3

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 100.0

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002 or 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.0

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 48.6%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 95.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.4:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_3 := th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot kx (sin ky))) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_3 (* th (/ (sin ky) (hypot (sin kx) (sin ky))))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.4)
       t_3
       (if (<= t_2 1e-6)
         (* (/ ky (hypot (sin kx) ky)) (sin th))
         (if (<= t_2 0.995) t_3 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_3 = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.4) {
		tmp = t_3;
	} else if (t_2 <= 1e-6) {
		tmp = (ky / hypot(sin(kx), ky)) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(kx, Math.sin(ky))) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_3 = th * (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.4) {
		tmp = t_3;
	} else if (t_2 <= 1e-6) {
		tmp = (ky / Math.hypot(Math.sin(kx), ky)) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(kx, math.sin(ky))) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_3 = th * (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky)))
	tmp = 0
	if t_2 <= -1.0:
		tmp = t_1
	elif t_2 <= -0.4:
		tmp = t_3
	elif t_2 <= 1e-6:
		tmp = (ky / math.hypot(math.sin(kx), ky)) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(kx, sin(ky))) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_3 = Float64(th * Float64(sin(ky) / hypot(sin(kx), sin(ky))))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.4)
		tmp = t_3;
	elseif (t_2 <= 1e-6)
		tmp = Float64(Float64(ky / hypot(sin(kx), ky)) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_3 = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.4)
		tmp = t_3;
	elseif (t_2 <= 1e-6)
		tmp = (ky / hypot(sin(kx), ky)) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.4], t$95$3, If[LessEqual[t$95$2, 1e-6], N[(N[(ky / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.3

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 100.0

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002 or 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.0

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 48.6%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 95.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.4:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ \mathbf{if}\;\frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}} \leq -0.4:\\ \;\;\;\;\sqrt{\frac{1}{t\_1}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0)))
   (if (<= (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))) -0.4)
     (* (sqrt (/ 1.0 t_1)) (* (sin ky) th))
     (* (/ ky (hypot (sin kx) ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double tmp;
	if ((sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)))) <= -0.4) {
		tmp = sqrt((1.0 / t_1)) * (sin(ky) * th);
	} else {
		tmp = (ky / hypot(sin(kx), ky)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(kx), 2.0)))) <= -0.4) {
		tmp = Math.sqrt((1.0 / t_1)) * (Math.sin(ky) * th);
	} else {
		tmp = (ky / Math.hypot(Math.sin(kx), ky)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	tmp = 0
	if (math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(kx), 2.0)))) <= -0.4:
		tmp = math.sqrt((1.0 / t_1)) * (math.sin(ky) * th)
	else:
		tmp = (ky / math.hypot(math.sin(kx), ky)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0)))) <= -0.4)
		tmp = Float64(sqrt(Float64(1.0 / t_1)) * Float64(sin(ky) * th));
	else
		tmp = Float64(Float64(ky / hypot(sin(kx), ky)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	tmp = 0.0;
	if ((sin(ky) / sqrt((t_1 + (sin(kx) ^ 2.0)))) <= -0.4)
		tmp = sqrt((1.0 / t_1)) * (sin(ky) * th);
	else
		tmp = (ky / hypot(sin(kx), ky)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.4], N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 29.7%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 95.9

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 76.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.4:\\ \;\;\;\;\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-6)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-6) {
		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) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-6) 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) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-6) {
		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) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-6:
		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 (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		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) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 97.0%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.0

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

   4. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-sin-a N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 35.4

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

   9. Applied rewrites 35.4%

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

   if 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-6)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-6) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-6) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-6) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-6:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 97.0

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 35.4

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

   7. Applied rewrites 35.4%

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

   if 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 5e-15)
   (* (/ (sin ky) (sin kx)) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 5e-15) {
		tmp = (sin(ky) / sin(kx)) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 5d-15) then
        tmp = (sin(ky) / sin(kx)) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 5e-15) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 5e-15:
		tmp = (math.sin(ky) / math.sin(kx)) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-15)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-15)
		tmp = (sin(ky) / sin(kx)) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.99999999999999999e-15

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 97.0

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 44.1%

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

   8. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.0

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

   10. Applied rewrites 23.0%

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

   if 4.99999999999999999e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + \mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 4e-12)
   (* (/ (sin ky) (hypot kx (sin ky))) (sin th))
   (*
    (/
     (sin ky)
     (sqrt
      (+ (- 0.5 (* (cos (* 2.0 ky)) 0.5)) (fma (cos (* 2.0 kx)) -0.5 0.5))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 4e-12) {
		tmp = (sin(ky) / hypot(kx, sin(ky))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * ky)) * 0.5)) + fma(cos((2.0 * kx)), -0.5, 0.5)))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 4e-12)
		tmp = Float64(Float64(sin(ky) / hypot(kx, sin(ky))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5)) + fma(cos(Float64(2.0 * kx)), -0.5, 0.5)))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 4e-12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 3.99999999999999992e-12

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.6

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.9

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

   3. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 99.8%

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

   if 3.99999999999999992e-12 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right) + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + \mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 5e-15)
   (* (/ th (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 5e-15) {
		tmp = (th / sin(kx)) * ky;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 5d-15) then
        tmp = (th / sin(kx)) * ky
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 5e-15) {
		tmp = (th / Math.sin(kx)) * ky;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 5e-15:
		tmp = (th / math.sin(kx)) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-15)
		tmp = Float64(Float64(th / 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) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-15)
		tmp = (th / sin(kx)) * ky;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.99999999999999999e-15

   1. Initial program 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 41.4

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

   5. Applied rewrites 41.4%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 22.2%

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

   if 4.99999999999999999e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (* (* (sqrt (/ 1.0 (fma (cos (* 2.0 kx)) -0.5 0.5))) ky) (sin th))
   (if (<= (sin kx) 5e-96) (sin th) (* (/ (sin ky) (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = (sqrt((1.0 / fma(cos((2.0 * kx)), -0.5, 0.5))) * ky) * sin(th);
	} else if (sin(kx) <= 5e-96) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(cos(Float64(2.0 * kx)), -0.5, 0.5))) * ky) * sin(th));
	elseif (sin(kx) <= 5e-96)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[(N[(N[Sqrt[N[(1.0 / N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-96], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 68.1

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \color{blue}{\left(2 \cdot kx\right)}, -0.5, 0.5\right)}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot ky\right)} \cdot \sin th \]

   if -0.0100000000000000002 < (sin.f64 kx) < 4.99999999999999995e-96

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 40.4

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

   5. Applied rewrites 40.4%

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

   if 4.99999999999999995e-96 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 59.4

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

   5. Applied rewrites 59.4%

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

Alternative 13: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-fma.f64 N/A

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

   11. count-2 N/A

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

   12. lower-cos.f64 N/A

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

   13. count-2 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 85.6

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

4. Applied rewrites 85.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. sqr-sin-a N/A

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

   15. lift-pow.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 14: 45.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (* (* (sqrt (/ 1.0 (fma (cos (* 2.0 kx)) -0.5 0.5))) ky) (sin th))
   (if (<= (sin kx) 5e-96) (sin th) (/ (sin th) (/ (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = (sqrt((1.0 / fma(cos((2.0 * kx)), -0.5, 0.5))) * ky) * sin(th);
	} else if (sin(kx) <= 5e-96) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(cos(Float64(2.0 * kx)), -0.5, 0.5))) * ky) * sin(th));
	elseif (sin(kx) <= 5e-96)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[(N[(N[Sqrt[N[(1.0 / N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-96], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 68.1

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \color{blue}{\left(2 \cdot kx\right)}, -0.5, 0.5\right)}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot ky\right)} \cdot \sin th \]

   if -0.0100000000000000002 < (sin.f64 kx) < 4.99999999999999995e-96

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 40.4

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

   5. Applied rewrites 40.4%

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

   if 4.99999999999999995e-96 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 88.0

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

   4. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-sin-a N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 52.9

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

   9. Applied rewrites 52.9%

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

Alternative 15: 45.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (* (* ky (sin th)) (sqrt (/ 1.0 (fma (cos (* 2.0 kx)) -0.5 0.5))))
   (if (<= (sin kx) 5e-96) (sin th) (/ (sin th) (/ (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(cos((2.0 * kx)), -0.5, 0.5)));
	} else if (sin(kx) <= 5e-96) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(cos(Float64(2.0 * kx)), -0.5, 0.5))));
	elseif (sin(kx) <= 5e-96)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-96], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 68.0

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot \left(\color{blue}{\sin th} \cdot ky\right) \]

   7. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot \left(\sin th \cdot ky\right)} \]

   if -0.0100000000000000002 < (sin.f64 kx) < 4.99999999999999995e-96

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 40.4

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

   5. Applied rewrites 40.4%

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

   if 4.99999999999999995e-96 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 88.0

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

   4. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-sin-a N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 52.9

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

   9. Applied rewrites 52.9%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 9.5e-5)
   (* th (/ (sin ky) (hypot (sin kx) (sin ky))))
   (* (/ ky (hypot (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 9.5e-5) {
		tmp = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (ky / hypot(sin(kx), ky)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 9.5e-5) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = (ky / Math.hypot(Math.sin(kx), ky)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 9.5e-5:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = (ky / math.hypot(math.sin(kx), ky)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 9.5e-5)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(ky / hypot(sin(kx), ky)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 9.5e-5)
		tmp = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	else
		tmp = (ky / hypot(sin(kx), ky)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 9.5e-5], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.5000000000000005e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 95.2

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 64.5%

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

   if 9.5000000000000005e-5 < th

   1. Initial program 95.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 95.4

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 61.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 66.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 95.2

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 99.7

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

3. Applied rewrites 99.7%

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

5. Taylor expanded in ky around 0

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

7. Applied rewrites 57.9%

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

8. Taylor expanded in ky around 0

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

10. Applied rewrites 69.8%

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

11. Final simplification 69.8%

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

Alternative 18: 23.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

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

Python

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

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 95.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 24.7

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

5. Applied rewrites 24.7%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 19: 13.6% accurate, 632.0× 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 95.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 43.8

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

5. Applied rewrites 43.8%

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

6. Taylor expanded in kx around 0

   \[\leadsto th \]
7. Step-by-step derivation

8. Applied rewrites 15.7%

   \[\leadsto th \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024272 
(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)))