Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.6%

Time: 11.8s

Alternatives: 26

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: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. 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. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 93.2

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

   8. lift-sqrt.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-hypot.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_5 := \left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{t\_4}\right)\\ \mathbf{if}\;t\_3 \leq -0.9995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{elif}\;t\_3 \leq 0.007:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th)))
        (t_3 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_4 (hypot (sin ky) (sin kx)))
        (t_5
         (*
          (- (sin th))
          (* (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (/ -1.0 t_4)))))
   (if (<= t_3 -0.9995)
     t_2
     (if (<= t_3 -0.05)
       (/
        1.0
        (/
         (*
          (/ (hypot (sin kx) (sin ky)) (sin ky))
          (fma (* th th) 0.16666666666666666 1.0))
         th))
       (if (<= t_3 0.007)
         t_5
         (if (<= t_3 0.99)
           (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_4)
           (if (<= t_3 2.0) t_2 t_5)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	double t_3 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_4 = hypot(sin(ky), sin(kx));
	double t_5 = -sin(th) * ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) * (-1.0 / t_4));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = t_2;
	} else if (t_3 <= -0.05) {
		tmp = 1.0 / (((hypot(sin(kx), sin(ky)) / sin(ky)) * fma((th * th), 0.16666666666666666, 1.0)) / th);
	} else if (t_3 <= 0.007) {
		tmp = t_5;
	} else if (t_3 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_4;
	} else if (t_3 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th))
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_4 = hypot(sin(ky), sin(kx))
	t_5 = Float64(Float64(-sin(th)) * Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) * Float64(-1.0 / t_4)))
	tmp = 0.0
	if (t_3 <= -0.9995)
		tmp = t_2;
	elseif (t_3 <= -0.05)
		tmp = Float64(1.0 / Float64(Float64(Float64(hypot(sin(kx), sin(ky)) / sin(ky)) * fma(Float64(th * th), 0.16666666666666666, 1.0)) / th));
	elseif (t_3 <= 0.007)
		tmp = t_5;
	elseif (t_3 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_4);
	elseif (t_3 <= 2.0)
		tmp = t_2;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sin[th], $MachinePrecision]) * N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(-1.0 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.9995], t$95$2, If[LessEqual[t$95$3, -0.05], N[(1.0 / N[(N[(N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.007], t$95$5, If[LessEqual[t$95$3, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 2.0], t$95$2, t$95$5]]]]]]]]]]

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))))) < -0.99950000000000006 or 0.98999999999999999 < (/.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))))) < 2

   1. Initial program 91.6%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.99950000000000006 < (/.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))))) < -0.050000000000000003

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 51.2%

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

   if -0.050000000000000003 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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. 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. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}\right)\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}\right)\right) \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky\right)\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky\right)\right) \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_5 := \left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{t\_4}\right)\\ \mathbf{if}\;t\_3 \leq -0.9995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_3 \leq 0.007:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th)))
        (t_3 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_4 (hypot (sin ky) (sin kx)))
        (t_5
         (*
          (- (sin th))
          (* (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (/ -1.0 t_4)))))
   (if (<= t_3 -0.9995)
     t_2
     (if (<= t_3 -0.05)
       (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)))
       (if (<= t_3 0.007)
         t_5
         (if (<= t_3 0.99)
           (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_4)
           (if (<= t_3 2.0) t_2 t_5)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	double t_3 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_4 = hypot(sin(ky), sin(kx));
	double t_5 = -sin(th) * ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) * (-1.0 / t_4));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = t_2;
	} else if (t_3 <= -0.05) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	} else if (t_3 <= 0.007) {
		tmp = t_5;
	} else if (t_3 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_4;
	} else if (t_3 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th))
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_4 = hypot(sin(ky), sin(kx))
	t_5 = Float64(Float64(-sin(th)) * Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) * Float64(-1.0 / t_4)))
	tmp = 0.0
	if (t_3 <= -0.9995)
		tmp = t_2;
	elseif (t_3 <= -0.05)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)));
	elseif (t_3 <= 0.007)
		tmp = t_5;
	elseif (t_3 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_4);
	elseif (t_3 <= 2.0)
		tmp = t_2;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sin[th], $MachinePrecision]) * N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(-1.0 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.9995], t$95$2, If[LessEqual[t$95$3, -0.05], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.007], t$95$5, If[LessEqual[t$95$3, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 2.0], t$95$2, t$95$5]]]]]]]]]]

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))))) < -0.99950000000000006 or 0.98999999999999999 < (/.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))))) < 2

   1. Initial program 91.6%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.99950000000000006 < (/.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))))) < -0.050000000000000003

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 50.0

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

   7. Applied rewrites 50.0%

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

   if -0.050000000000000003 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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. 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. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}\right)\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}\right)\right) \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky\right)\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky\right)\right) \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ t_4 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ t_5 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_3 \leq -0.9995:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_3 \leq 0.007:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_5}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{t\_5}}{\frac{1}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2))))
        (t_4 (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th)))
        (t_5 (hypot (sin ky) (sin kx))))
   (if (<= t_3 -0.9995)
     t_4
     (if (<= t_3 -0.05)
       (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)))
       (if (<= t_3 0.007)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_2))) (sin th))
         (if (<= t_3 0.99)
           (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_5)
           (if (<= t_3 2.0) t_4 (/ (/ (sin th) t_5) (/ 1.0 ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double t_4 = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	double t_5 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = t_4;
	} else if (t_3 <= -0.05) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	} else if (t_3 <= 0.007) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_2))) * sin(th);
	} else if (t_3 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_5;
	} else if (t_3 <= 2.0) {
		tmp = t_4;
	} else {
		tmp = (sin(th) / t_5) / (1.0 / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	t_4 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th))
	t_5 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_3 <= -0.9995)
		tmp = t_4;
	elseif (t_3 <= -0.05)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)));
	elseif (t_3 <= 0.007)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_2))) * sin(th));
	elseif (t_3 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_5);
	elseif (t_3 <= 2.0)
		tmp = t_4;
	else
		tmp = Float64(Float64(sin(th) / t_5) / Float64(1.0 / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -0.9995], t$95$4, If[LessEqual[t$95$3, -0.05], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.007], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$3, 2.0], t$95$4, N[(N[(N[Sin[th], $MachinePrecision] / t$95$5), $MachinePrecision] / N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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))))) < -0.99950000000000006 or 0.98999999999999999 < (/.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))))) < 2

   1. Initial program 91.6%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.99950000000000006 < (/.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))))) < -0.050000000000000003

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 50.0

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

   7. Applied rewrites 50.0%

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

   if -0.050000000000000003 < (/.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))))) < 0.00700000000000000015

   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

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

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

   if 2 < (/.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 2.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\color{blue}{\frac{1}{ky}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\frac{\sin th}{t\_1}}{\frac{1}{ky}}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ t_5 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_5 \leq -0.9995:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_5 \leq 0.007:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_1}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (/ (sin th) t_1) (/ 1.0 ky)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th)))
        (t_5 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0))))))
   (if (<= t_5 -0.9995)
     t_4
     (if (<= t_5 -0.01)
       (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)))
       (if (<= t_5 0.007)
         t_2
         (if (<= t_5 0.99)
           (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_1)
           (if (<= t_5 2.0) t_4 t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(th) / t_1) / (1.0 / ky);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	double t_5 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_5 <= -0.9995) {
		tmp = t_4;
	} else if (t_5 <= -0.01) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	} else if (t_5 <= 0.007) {
		tmp = t_2;
	} else if (t_5 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_1;
	} else if (t_5 <= 2.0) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(th) / t_1) / Float64(1.0 / ky))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th))
	t_5 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9995)
		tmp = t_4;
	elseif (t_5 <= -0.01)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)));
	elseif (t_5 <= 0.007)
		tmp = t_2;
	elseif (t_5 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_1);
	elseif (t_5 <= 2.0)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.9995], t$95$4, If[LessEqual[t$95$5, -0.01], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.007], t$95$2, If[LessEqual[t$95$5, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$5, 2.0], t$95$4, t$95$2]]]]]]]]]]

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))))) < -0.99950000000000006 or 0.98999999999999999 < (/.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))))) < 2

   1. Initial program 91.6%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.99950000000000006 < (/.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))))) < -0.0100000000000000002

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 48.5

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

   7. Applied rewrites 48.5%

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

   if -0.0100000000000000002 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\frac{\sin th}{t\_1}}{\frac{1}{ky}}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)}\\ \mathbf{elif}\;t\_4 \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_4 \leq 0.007:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{t\_3}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (/ (sin th) t_1) (/ 1.0 ky)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0))))))
   (if (<= t_4 -0.9995)
     (/
      (* (sin ky) (sin th))
      (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
     (if (<= t_4 -0.01)
       (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)))
       (if (<= t_4 0.007)
         t_2
         (if (<= t_4 0.99)
           (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_1)
           (if (<= t_4 2.0)
             (/ (sin th) (fma (/ 0.5 t_3) (* kx kx) 1.0))
             t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(th) / t_1) / (1.0 / ky);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_4 <= -0.9995) {
		tmp = (sin(ky) * sin(th)) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx));
	} else if (t_4 <= -0.01) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	} else if (t_4 <= 0.007) {
		tmp = t_2;
	} else if (t_4 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_1;
	} else if (t_4 <= 2.0) {
		tmp = sin(th) / fma((0.5 / t_3), (kx * kx), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(th) / t_1) / Float64(1.0 / ky))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.9995)
		tmp = Float64(Float64(sin(ky) * sin(th)) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx)));
	elseif (t_4 <= -0.01)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)));
	elseif (t_4 <= 0.007)
		tmp = t_2;
	elseif (t_4 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_1);
	elseif (t_4 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(0.5 / t_3), Float64(kx * kx), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.01], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.007], t$95$2, If[LessEqual[t$95$4, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(0.5 / t$95$3), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 5 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))))) < -0.99950000000000006

   1. Initial program 85.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.6

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \]

      6. lower-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

   if -0.99950000000000006 < (/.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))))) < -0.0100000000000000002

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 48.5

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

   7. Applied rewrites 48.5%

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

   if -0.0100000000000000002 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

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

   if 0.98999999999999999 < (/.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))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.4

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

   7. Applied rewrites 93.4%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_2 \leq -0.9995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_2 \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (/
          (* (sin ky) (sin th))
          (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin ky) (sin kx))))
   (if (<= t_2 -0.9995)
     t_1
     (if (<= t_2 -0.01)
       (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)))
       (if (<= t_2 0.007)
         (/ (* ky (sin th)) t_3)
         (if (<= t_2 0.99)
           (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_3)
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * sin(th)) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx));
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_2 <= -0.9995) {
		tmp = t_1;
	} else if (t_2 <= -0.01) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	} else if (t_2 <= 0.007) {
		tmp = (ky * sin(th)) / t_3;
	} else if (t_2 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * sin(th)) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_2 <= -0.9995)
		tmp = t_1;
	elseif (t_2 <= -0.01)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)));
	elseif (t_2 <= 0.007)
		tmp = Float64(Float64(ky * sin(th)) / t_3);
	elseif (t_2 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_3);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $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[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -0.9995], t$95$1, If[LessEqual[t$95$2, -0.01], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.007], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$3), $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))))) < -0.99950000000000006 or 0.98999999999999999 < (/.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 83.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 81.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \]

      6. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if -0.99950000000000006 < (/.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))))) < -0.0100000000000000002

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 48.5

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

   7. Applied rewrites 48.5%

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

   if -0.0100000000000000002 < (/.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))))) < 0.00700000000000000015

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.01:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.007:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{t\_3}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* ky (sin th)) t_1))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0))))))
   (if (<= t_4 -0.01)
     (/ (* (sin ky) th) t_1)
     (if (<= t_4 0.007)
       t_2
       (if (<= t_4 0.99)
         (/ (* (* (fma (* th th) -0.16666666666666666 1.0) (sin ky)) th) t_1)
         (if (<= t_4 2.0)
           (/ (sin th) (fma (/ 0.5 t_3) (* kx kx) 1.0))
           t_2))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (ky * sin(th)) / t_1;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_4 <= -0.01) {
		tmp = (sin(ky) * th) / t_1;
	} else if (t_4 <= 0.007) {
		tmp = t_2;
	} else if (t_4 <= 0.99) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_1;
	} else if (t_4 <= 2.0) {
		tmp = sin(th) / fma((0.5 / t_3), (kx * kx), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(ky * sin(th)) / t_1)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.01)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (t_4 <= 0.007)
		tmp = t_2;
	elseif (t_4 <= 0.99)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * sin(ky)) * th) / t_1);
	elseif (t_4 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(0.5 / t_3), Float64(kx * kx), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.007], t$95$2, If[LessEqual[t$95$4, 0.99], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(0.5 / t$95$3), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

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))))) < -0.0100000000000000002

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 43.0

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

   7. Applied rewrites 43.0%

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

   if -0.0100000000000000002 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if 0.00700000000000000015 < (/.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))))) < 0.98999999999999999

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin ky + \left(\frac{-1}{6} \cdot \sin ky\right) \cdot {th}^{2}\right) \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\frac{-1}{6} \cdot \left(\sin ky \cdot {th}^{2}\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin ky + \frac{-1}{6} \cdot \color{blue}{\left({th}^{2} \cdot \sin ky\right)}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky\right)} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 72.9

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

   7. Applied rewrites 72.9%

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

   if 0.98999999999999999 < (/.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))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.4

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

   7. Applied rewrites 93.4%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \sin ky\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot th\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_3 := \frac{ky \cdot \sin th}{t\_2}\\ t_4 := {\sin ky}^{2}\\ t_5 := \frac{\sin ky}{\sqrt{t\_4 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_5 \leq -0.01:\\ \;\;\;\;\frac{t\_1}{t\_2}\\ \mathbf{elif}\;t\_5 \leq 0.007:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 0.995:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{t\_1}}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{t\_4}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) th))
        (t_2 (hypot (sin ky) (sin kx)))
        (t_3 (/ (* ky (sin th)) t_2))
        (t_4 (pow (sin ky) 2.0))
        (t_5 (/ (sin ky) (sqrt (+ t_4 (pow (sin kx) 2.0))))))
   (if (<= t_5 -0.01)
     (/ t_1 t_2)
     (if (<= t_5 0.007)
       t_3
       (if (<= t_5 0.995)
         (/ 1.0 (/ (hypot (sin kx) (sin ky)) t_1))
         (if (<= t_5 2.0)
           (/ (sin th) (fma (/ 0.5 t_4) (* kx kx) 1.0))
           t_3))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * th;
	double t_2 = hypot(sin(ky), sin(kx));
	double t_3 = (ky * sin(th)) / t_2;
	double t_4 = pow(sin(ky), 2.0);
	double t_5 = sin(ky) / sqrt((t_4 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_5 <= -0.01) {
		tmp = t_1 / t_2;
	} else if (t_5 <= 0.007) {
		tmp = t_3;
	} else if (t_5 <= 0.995) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / t_1);
	} else if (t_5 <= 2.0) {
		tmp = sin(th) / fma((0.5 / t_4), (kx * kx), 1.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * th)
	t_2 = hypot(sin(ky), sin(kx))
	t_3 = Float64(Float64(ky * sin(th)) / t_2)
	t_4 = sin(ky) ^ 2.0
	t_5 = Float64(sin(ky) / sqrt(Float64(t_4 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.01)
		tmp = Float64(t_1 / t_2);
	elseif (t_5 <= 0.007)
		tmp = t_3;
	elseif (t_5 <= 0.995)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / t_1));
	elseif (t_5 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(0.5 / t_4), Float64(kx * kx), 1.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.01], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 0.007], t$95$3, If[LessEqual[t$95$5, 0.995], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(0.5 / t$95$4), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

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))))) < -0.0100000000000000002

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 43.0

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

   7. Applied rewrites 43.0%

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

   if -0.0100000000000000002 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if 0.00700000000000000015 < (/.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 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if 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))))) < 2

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := \frac{ky \cdot \sin th}{t\_1}\\ t_4 := {\sin ky}^{2}\\ t_5 := \frac{\sin ky}{\sqrt{t\_4 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_5 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 0.007:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{t\_4}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (/ (* ky (sin th)) t_1))
        (t_4 (pow (sin ky) 2.0))
        (t_5 (/ (sin ky) (sqrt (+ t_4 (pow (sin kx) 2.0))))))
   (if (<= t_5 -0.01)
     t_2
     (if (<= t_5 0.007)
       t_3
       (if (<= t_5 0.995)
         t_2
         (if (<= t_5 2.0)
           (/ (sin th) (fma (/ 0.5 t_4) (* kx kx) 1.0))
           t_3))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = (ky * sin(th)) / t_1;
	double t_4 = pow(sin(ky), 2.0);
	double t_5 = sin(ky) / sqrt((t_4 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_5 <= -0.01) {
		tmp = t_2;
	} else if (t_5 <= 0.007) {
		tmp = t_3;
	} else if (t_5 <= 0.995) {
		tmp = t_2;
	} else if (t_5 <= 2.0) {
		tmp = sin(th) / fma((0.5 / t_4), (kx * kx), 1.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = Float64(Float64(ky * sin(th)) / t_1)
	t_4 = sin(ky) ^ 2.0
	t_5 = Float64(sin(ky) / sqrt(Float64(t_4 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.01)
		tmp = t_2;
	elseif (t_5 <= 0.007)
		tmp = t_3;
	elseif (t_5 <= 0.995)
		tmp = t_2;
	elseif (t_5 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(0.5 / t_4), Float64(kx * kx), 1.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.01], t$95$2, If[LessEqual[t$95$5, 0.007], t$95$3, If[LessEqual[t$95$5, 0.995], t$95$2, If[LessEqual[t$95$5, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(0.5 / t$95$4), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

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))))) < -0.0100000000000000002 or 0.00700000000000000015 < (/.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 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.6

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if -0.0100000000000000002 < (/.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))))) < 0.00700000000000000015 or 2 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if 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))))) < 2

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{{\sin ky}^{2}}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_2 -0.05)
     t_1
     (if (<= t_2 2e-49)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_2 0.99) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= 2e-49) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 0.99) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= 2e-49) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_2 <= 0.99) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_2 <= -0.05:
		tmp = t_1
	elif t_2 <= 2e-49:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_2 <= 0.99:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= 2e-49)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 0.99)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= 2e-49)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_2 <= 0.99)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $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]}, If[LessEqual[t$95$2, -0.05], t$95$1, If[LessEqual[t$95$2, 2e-49], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99], t$95$1, N[Sin[th], $MachinePrecision]]]]]]

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))))) < -0.050000000000000003 or 1.99999999999999987e-49 < (/.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))))) < 0.98999999999999999

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 53.6

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

   7. Applied rewrites 53.6%

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

   if -0.050000000000000003 < (/.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))))) < 1.99999999999999987e-49

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if 0.98999999999999999 < (/.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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.99:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right) \cdot th, ky \cdot ky, th\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 2e-49)
     (* (/ (sin ky) (sin kx)) (sin th))
     (if (<= t_1 0.01)
       (/
        1.0
        (/
         (hypot (sin kx) (sin ky))
         (*
          (fma
           (* (fma 0.008333333333333333 (* ky ky) -0.16666666666666666) th)
           (* ky ky)
           th)
          ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= 2e-49) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.01) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (fma((fma(0.008333333333333333, (ky * ky), -0.16666666666666666) * th), (ky * ky), th) * ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-49)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.01)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(fma(Float64(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666) * th), Float64(ky * ky), th) * ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 2e-49], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + th), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

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))))) < 1.99999999999999987e-49

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 32.8

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

   5. Applied rewrites 32.8%

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

   if 1.99999999999999987e-49 < (/.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))))) < 0.0100000000000000002

   1. Initial program 98.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 75.2

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky \cdot \color{blue}{\left(th + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot th + \frac{1}{120} \cdot \left({ky}^{2} \cdot th\right)\right)\right)}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.1%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(th \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, th\right) \cdot \color{blue}{ky}}} \]

   if 0.0100000000000000002 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right) \cdot th, ky \cdot ky, th\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot th\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 2e-49)
     (* (/ (sin ky) (sin kx)) (sin th))
     (if (<= t_1 0.01)
       (/
        1.0
        (/
         (hypot (sin kx) (sin ky))
         (* (* (fma -0.16666666666666666 (* ky ky) 1.0) th) ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= 2e-49) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.01) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / ((fma(-0.16666666666666666, (ky * ky), 1.0) * th) * ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-49)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.01)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * th) * ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 2e-49], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

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))))) < 1.99999999999999987e-49

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 32.8

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

   5. Applied rewrites 32.8%

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

   if 1.99999999999999987e-49 < (/.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))))) < 0.0100000000000000002

   1. Initial program 98.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 75.2

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 74.6%

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

   if 0.0100000000000000002 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot th\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.3)
   (* (/ (sin 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)))) <= 0.3) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.3d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.3) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.3:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = Float64(Float64(sin(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)))) <= 0.3)
		tmp = (sin(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], 0.3], N[(N[(N[Sin[ky], $MachinePrecision] / 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))))) < 0.299999999999999989

   1. Initial program 95.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   if 0.299999999999999989 < (/.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 86.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.2

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

   5. Applied rewrites 66.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.007)
   (* (/ 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)))) <= 0.007) {
		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)))) <= 0.007d0) 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)))) <= 0.007) {
		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)))) <= 0.007:
		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)))) <= 0.007)
		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)))) <= 0.007)
		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], 0.007], 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))))) < 0.00700000000000000015

   1. Initial program 95.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.2

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

   5. Applied rewrites 33.2%

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

   if 0.00700000000000000015 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.007)
   (/ (* ky (sin th)) (sin kx))
   (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)))) <= 0.007) {
		tmp = (ky * sin(th)) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.007d0) then
        tmp = (ky * sin(th)) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.007) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.007:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.007)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.007)
		tmp = (ky * sin(th)) / sin(kx);
	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], 0.007], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $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))))) < 0.00700000000000000015

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 32.7

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

   7. Applied rewrites 32.7%

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

   if 0.00700000000000000015 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.008333333333333333}{ky} \cdot kx, kx, \frac{-0.16666666666666666}{ky}\right), kx \cdot kx, \frac{1}{ky}\right) \cdot kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.007)
   (/
    1.0
    (/
     (*
      (fma
       (fma (* (/ 0.008333333333333333 ky) kx) kx (/ -0.16666666666666666 ky))
       (* kx kx)
       (/ 1.0 ky))
      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)))) <= 0.007) {
		tmp = 1.0 / ((fma(fma(((0.008333333333333333 / ky) * kx), kx, (-0.16666666666666666 / ky)), (kx * kx), (1.0 / ky)) * kx) / sin(th));
	} else {
		tmp = 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)))) <= 0.007)
		tmp = Float64(1.0 / Float64(Float64(fma(fma(Float64(Float64(0.008333333333333333 / ky) * kx), kx, Float64(-0.16666666666666666 / ky)), Float64(kx * kx), Float64(1.0 / ky)) * kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return 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], 0.007], N[(1.0 / N[(N[(N[(N[(N[(N[(0.008333333333333333 / ky), $MachinePrecision] * kx), $MachinePrecision] * kx + N[(-0.16666666666666666 / ky), $MachinePrecision]), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] / N[Sin[th], $MachinePrecision]), $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))))) < 0.00700000000000000015

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\frac{kx \cdot \color{blue}{\left({kx}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{kx}^{2}}{ky} - \frac{1}{6} \cdot \frac{1}{ky}\right) + \frac{1}{ky}\right)}}{\sin th}} \]
   9. Step-by-step derivation

   10. Applied rewrites 20.4%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.008333333333333333}{ky} \cdot kx, kx, \frac{-0.16666666666666666}{ky}\right), kx \cdot kx, \frac{1}{ky}\right) \cdot \color{blue}{kx}}{\sin th}} \]

   if 0.00700000000000000015 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.008333333333333333}{ky} \cdot kx, kx, \frac{-0.16666666666666666}{ky}\right), kx \cdot kx, \frac{1}{ky}\right) \cdot kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{kx \cdot kx}{ky}, -0.16666666666666666, \frac{1}{ky}\right) \cdot kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.007)
   (/
    1.0
    (/ (* (fma (/ (* kx kx) ky) -0.16666666666666666 (/ 1.0 ky)) 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)))) <= 0.007) {
		tmp = 1.0 / ((fma(((kx * kx) / ky), -0.16666666666666666, (1.0 / ky)) * kx) / sin(th));
	} else {
		tmp = 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)))) <= 0.007)
		tmp = Float64(1.0 / Float64(Float64(fma(Float64(Float64(kx * kx) / ky), -0.16666666666666666, Float64(1.0 / ky)) * kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return 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], 0.007], N[(1.0 / N[(N[(N[(N[(N[(kx * kx), $MachinePrecision] / ky), $MachinePrecision] * -0.16666666666666666 + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] / N[Sin[th], $MachinePrecision]), $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))))) < 0.00700000000000000015

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 20.3%

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

   if 0.00700000000000000015 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{kx \cdot kx}{ky}, -0.16666666666666666, \frac{1}{ky}\right) \cdot kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{\frac{kx}{ky}}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.007)
   (/ 1.0 (/ (/ kx ky) (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)))) <= 0.007) {
		tmp = 1.0 / ((kx / ky) / 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)))) <= 0.007d0) then
        tmp = 1.0d0 / ((kx / ky) / 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)))) <= 0.007) {
		tmp = 1.0 / ((kx / ky) / 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)))) <= 0.007:
		tmp = 1.0 / ((kx / ky) / 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)))) <= 0.007)
		tmp = Float64(1.0 / Float64(Float64(kx / ky) / 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)))) <= 0.007)
		tmp = 1.0 / ((kx / ky) / 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], 0.007], N[(1.0 / N[(N[(kx / ky), $MachinePrecision] / N[Sin[th], $MachinePrecision]), $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))))) < 0.00700000000000000015

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 20.5%

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

   if 0.00700000000000000015 < (/.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 87.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{\frac{kx}{ky}}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{ky}}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.005)
   (/ 1.0 (/ (/ (sin kx) ky) 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)))) <= 0.005) {
		tmp = 1.0 / ((sin(kx) / ky) / 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)))) <= 0.005d0) then
        tmp = 1.0d0 / ((sin(kx) / ky) / 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)))) <= 0.005) {
		tmp = 1.0 / ((Math.sin(kx) / ky) / 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)))) <= 0.005:
		tmp = 1.0 / ((math.sin(kx) / ky) / 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)))) <= 0.005)
		tmp = Float64(1.0 / Float64(Float64(sin(kx) / ky) / 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)))) <= 0.005)
		tmp = 1.0 / ((sin(kx) / ky) / 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], 0.005], N[(1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision] / 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))))) < 0.0050000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}{\sin th}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}{\sin th}}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}{\sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th \cdot \sin ky} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{th \cdot \sin ky}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{th \cdot \sin ky}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{th \cdot \sin ky}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{th \cdot \sin ky}} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{th \cdot \sin ky}} \]

      6. lower-hypot.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot \sin ky}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\color{blue}{\sin kx}, \sin ky\right)}{th \cdot \sin ky}} \]

      8. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)}{th \cdot \sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\sin ky \cdot th}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\sin ky \cdot th}}} \]

      11. lower-sin.f64 45.2

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\sin ky} \cdot th}} \]

   7. Applied rewrites 45.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{ky \cdot th}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 20.3%

       \[\leadsto \frac{1}{\frac{\frac{\sin kx}{ky}}{\color{blue}{th}}} \]

   if 0.0050000000000000001 < (/.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 88.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{ky}}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-44}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-44)
   (* (pow th 3.0) -0.16666666666666666)
   (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-44) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} 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-44) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    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-44) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} 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-44:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	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-44)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	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-44)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	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-44], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $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))))) < 9.99999999999999953e-45

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.1%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 12.5%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 9.99999999999999953e-45 < (/.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 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 54.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-44}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin th) (hypot (sin ky) (sin kx))) (sin ky)))

C

double code(double kx, double ky, double th) {
	return (sin(th) / hypot(sin(ky), sin(kx))) * sin(ky);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(th) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(ky);
}

Python

def code(kx, ky, th):
	return (math.sin(th) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(ky)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(th) / hypot(sin(ky), sin(kx))) * sin(ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(th) / hypot(sin(ky), sin(kx))) * sin(ky);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. 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. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   7. lower-/.f64 93.1

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   10. +-commutative N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

   12. unpow2 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

   14. unpow2 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

   15. lower-hypot.f64 99.6

       \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
5. Add Preprocessing

Alternative 23: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0009:\\ \;\;\;\;\left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0009)
   (*
    (- (sin th))
    (*
     (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
     (/ -1.0 (hypot (sin ky) (sin kx)))))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* (- 1.0 (cos (* 2.0 kx))) 2.0)))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0009) {
		tmp = -sin(th) * ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) * (-1.0 / hypot(sin(ky), sin(kx))));
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, ((1.0 - cos((2.0 * kx))) * 2.0))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0009)
		tmp = Float64(Float64(-sin(th)) * Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) * Float64(-1.0 / hypot(sin(ky), sin(kx)))));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0009], N[((-N[Sin[th], $MachinePrecision]) * N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 8.9999999999999998e-4

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      3. frac-2neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}{\mathsf{neg}\left(\sin ky\right)}}} \cdot \sin th \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \sin th \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\sin ky\right)\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}\right)\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}\right)\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}\right)\right) \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky\right)\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky\right)\right) \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 70.1

         \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky\right)\right) \cdot \sin th \]

   7. Applied rewrites 70.1%

      \[\leadsto \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(-\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}\right)\right) \cdot \sin th \]

   if 8.9999999999999998e-4 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0009:\\ \;\;\;\;\left(-\sin th\right) \cdot \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 23.9% 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 93.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 20.6

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 20.6%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 25: 13.1% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{th}} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (/ 1.0 (/ 1.0 th)))

C

double code(double kx, double ky, double th) {
	return 1.0 / (1.0 / 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 = 1.0d0 / (1.0d0 / th)
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 / (1.0 / th);
}

Python

def code(kx, ky, th):
	return 1.0 / (1.0 / th)

Julia

function code(kx, ky, th)
	return Float64(1.0 / Float64(1.0 / th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 / (1.0 / th);
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 / N[(1.0 / th), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. 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. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}{\sin th}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}{\sin th}}} \]

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}{\sin th}}} \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{th \cdot \sin ky} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
6. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{th \cdot \sin ky}}} \]

   2. *-lft-identity N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{th \cdot \sin ky}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{th \cdot \sin ky}}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{th \cdot \sin ky}} \]

   5. unpow2 N/A

      \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{th \cdot \sin ky}} \]

   6. lower-hypot.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot \sin ky}} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\color{blue}{\sin kx}, \sin ky\right)}{th \cdot \sin ky}} \]

   8. lower-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)}{th \cdot \sin ky}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\sin ky \cdot th}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\sin ky \cdot th}}} \]

   11. lower-sin.f64 46.6

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\sin ky} \cdot th}} \]

7. Applied rewrites 46.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}} \]

8. Taylor expanded in kx around 0

   \[\leadsto \frac{1}{\frac{1}{\color{blue}{th}}} \]
9. Step-by-step derivation

10. Applied rewrites 11.9%

    \[\leadsto \frac{1}{\frac{1}{\color{blue}{th}}} \]
11. Add Preprocessing

Alternative 26: 12.9% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(th \cdot th\right) \cdot -0.16666666666666666, th, th\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (fma (* (* th th) -0.16666666666666666) th th))

C

double code(double kx, double ky, double th) {
	return fma(((th * th) * -0.16666666666666666), th, th);
}

Julia

function code(kx, ky, th)
	return fma(Float64(Float64(th * th) * -0.16666666666666666), th, th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th + th), $MachinePrecision]

Derivation

1. Initial program 93.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 20.6

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 20.6%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 11.6%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]

11. Final simplification 11.6%

    \[\leadsto \mathsf{fma}\left(\left(th \cdot th\right) \cdot -0.16666666666666666, th, th\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024278 
(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)))