Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.6%

Time: 10.3s

Alternatives: 18

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

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

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 92.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. 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 92.7

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

   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 2: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + {\sin kx}^{2}}}\\ t_5 := \frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{if}\;t\_4 \leq -0.998:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -0.02:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.98:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (- ky) (/ -1.0 (hypot (sin ky) (sin kx)))) (sin th)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th)))
        (t_4 (/ (sin ky) (sqrt (+ t_2 (pow (sin kx) 2.0)))))
        (t_5
         (/
          1.0
          (/
           (*
            (/ (hypot (sin kx) (sin ky)) (sin ky))
            (fma (* th th) 0.16666666666666666 1.0))
           th))))
   (if (<= t_4 -0.998)
     t_3
     (if (<= t_4 -0.02)
       t_5
       (if (<= t_4 5e-14)
         t_1
         (if (<= t_4 0.98) t_5 (if (<= t_4 2.0) t_3 t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (-ky * (-1.0 / hypot(sin(ky), sin(kx)))) * sin(th);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	double t_4 = sin(ky) / sqrt((t_2 + pow(sin(kx), 2.0)));
	double t_5 = 1.0 / (((hypot(sin(kx), sin(ky)) / sin(ky)) * fma((th * th), 0.16666666666666666, 1.0)) / th);
	double tmp;
	if (t_4 <= -0.998) {
		tmp = t_3;
	} else if (t_4 <= -0.02) {
		tmp = t_5;
	} else if (t_4 <= 5e-14) {
		tmp = t_1;
	} else if (t_4 <= 0.98) {
		tmp = t_5;
	} else if (t_4 <= 2.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-ky) * Float64(-1.0 / hypot(sin(ky), sin(kx)))) * sin(th))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th))
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(kx) ^ 2.0))))
	t_5 = Float64(1.0 / Float64(Float64(Float64(hypot(sin(kx), sin(ky)) / sin(ky)) * fma(Float64(th * th), 0.16666666666666666, 1.0)) / th))
	tmp = 0.0
	if (t_4 <= -0.998)
		tmp = t_3;
	elseif (t_4 <= -0.02)
		tmp = t_5;
	elseif (t_4 <= 5e-14)
		tmp = t_1;
	elseif (t_4 <= 0.98)
		tmp = t_5;
	elseif (t_4 <= 2.0)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[((-ky) * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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$4, -0.998], t$95$3, If[LessEqual[t$95$4, -0.02], t$95$5, If[LessEqual[t$95$4, 5e-14], t$95$1, If[LessEqual[t$95$4, 0.98], t$95$5, If[LessEqual[t$95$4, 2.0], t$95$3, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998 or 0.97999999999999998 < (/.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 92.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

   if -0.998 < (/.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))))) < -0.0200000000000000004 or 5.0000000000000002e-14 < (/.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))))) < 0.97999999999999998

   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 99.0%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 43.5%

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

   if -0.0200000000000000004 < (/.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))))) < 5.0000000000000002e-14 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 88.6%

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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_4 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_5 := \left(\left(-ky\right) \cdot t\_4\right) \cdot \sin th\\ \mathbf{if}\;t\_3 \leq -0.998:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -0.02:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 0.98:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

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 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_5 (* (* (- ky) t_4) (sin th))))
   (if (<= t_3 -0.998)
     t_2
     (if (<= t_3 -0.02)
       (* (* (- th) (sin ky)) t_4)
       (if (<= t_3 5e-14)
         t_5
         (if (<= t_3 0.98)
           (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)))
           (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 = -1.0 / hypot(sin(ky), sin(kx));
	double t_5 = (-ky * t_4) * sin(th);
	double tmp;
	if (t_3 <= -0.998) {
		tmp = t_2;
	} else if (t_3 <= -0.02) {
		tmp = (-th * sin(ky)) * t_4;
	} else if (t_3 <= 5e-14) {
		tmp = t_5;
	} else if (t_3 <= 0.98) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	} else if (t_3 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	double t_3 = Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(kx), 2.0)));
	double t_4 = -1.0 / Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_5 = (-ky * t_4) * Math.sin(th);
	double tmp;
	if (t_3 <= -0.998) {
		tmp = t_2;
	} else if (t_3 <= -0.02) {
		tmp = (-th * Math.sin(ky)) * t_4;
	} else if (t_3 <= 5e-14) {
		tmp = t_5;
	} else if (t_3 <= 0.98) {
		tmp = 1.0 / (Math.hypot(Math.sin(kx), Math.sin(ky)) / (Math.sin(ky) * th));
	} else if (t_3 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	t_3 = math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(kx), 2.0)))
	t_4 = -1.0 / math.hypot(math.sin(ky), math.sin(kx))
	t_5 = (-ky * t_4) * math.sin(th)
	tmp = 0
	if t_3 <= -0.998:
		tmp = t_2
	elif t_3 <= -0.02:
		tmp = (-th * math.sin(ky)) * t_4
	elif t_3 <= 5e-14:
		tmp = t_5
	elif t_3 <= 0.98:
		tmp = 1.0 / (math.hypot(math.sin(kx), math.sin(ky)) / (math.sin(ky) * th))
	elif 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 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_5 = Float64(Float64(Float64(-ky) * t_4) * sin(th))
	tmp = 0.0
	if (t_3 <= -0.998)
		tmp = t_2;
	elseif (t_3 <= -0.02)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * t_4);
	elseif (t_3 <= 5e-14)
		tmp = t_5;
	elseif (t_3 <= 0.98)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)));
	elseif (t_3 <= 2.0)
		tmp = t_2;
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	t_3 = sin(ky) / sqrt((t_1 + (sin(kx) ^ 2.0)));
	t_4 = -1.0 / hypot(sin(ky), sin(kx));
	t_5 = (-ky * t_4) * sin(th);
	tmp = 0.0;
	if (t_3 <= -0.998)
		tmp = t_2;
	elseif (t_3 <= -0.02)
		tmp = (-th * sin(ky)) * t_4;
	elseif (t_3 <= 5e-14)
		tmp = t_5;
	elseif (t_3 <= 0.98)
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(ky) * th));
	elseif (t_3 <= 2.0)
		tmp = t_2;
	else
		tmp = t_5;
	end
	tmp_2 = 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[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[((-ky) * t$95$4), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.998], t$95$2, If[LessEqual[t$95$3, -0.02], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 5e-14], t$95$5, If[LessEqual[t$95$3, 0.98], 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, 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))))) < -0.998 or 0.97999999999999998 < (/.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 92.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

   if -0.998 < (/.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))))) < -0.0200000000000000004

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 44.9

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

   7. Applied rewrites 44.9%

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

   if -0.0200000000000000004 < (/.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))))) < 5.0000000000000002e-14 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 88.6%

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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 5.0000000000000002e-14 < (/.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))))) < 0.97999999999999998

   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 98.9%

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

   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 42.0

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

   7. Applied rewrites 42.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \left(\left(-th\right) \cdot \sin ky\right) \cdot t\_1\\ t_3 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.9999998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_2 (* (* (- th) (sin ky)) t_1))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_3 -0.02)
     t_2
     (if (<= t_3 0.2)
       (* (* (- ky) (sin th)) t_1)
       (if (<= t_3 0.9999998) t_2 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -1.0 / hypot(sin(ky), sin(kx));
	double t_2 = (-th * sin(ky)) * t_1;
	double t_3 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_3 <= -0.02) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = (-ky * sin(th)) * t_1;
	} else if (t_3 <= 0.9999998) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -1.0 / Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = (-th * Math.sin(ky)) * t_1;
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_3 <= -0.02) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = (-ky * Math.sin(th)) * t_1;
	} else if (t_3 <= 0.9999998) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -1.0 / math.hypot(math.sin(ky), math.sin(kx))
	t_2 = (-th * math.sin(ky)) * t_1
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_3 <= -0.02:
		tmp = t_2
	elif t_3 <= 0.2:
		tmp = (-ky * math.sin(th)) * t_1
	elif t_3 <= 0.9999998:
		tmp = t_2
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_2 = Float64(Float64(Float64(-th) * sin(ky)) * t_1)
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.02)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(Float64(-ky) * sin(th)) * t_1);
	elseif (t_3 <= 0.9999998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -1.0 / hypot(sin(ky), sin(kx));
	t_2 = (-th * sin(ky)) * t_1;
	t_3 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.02)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = (-ky * sin(th)) * t_1;
	elseif (t_3 <= 0.9999998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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$3, -0.02], t$95$2, If[LessEqual[t$95$3, 0.2], N[(N[((-ky) * N[Sin[th], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.9999998], t$95$2, 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))))) < -0.0200000000000000004 or 0.20000000000000001 < (/.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))))) < 0.999999799999999994

   1. Initial program 92.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 91.9

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 42.8

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

   7. Applied rewrites 42.8%

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

   if -0.0200000000000000004 < (/.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))))) < 0.20000000000000001

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 94.8

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   7. Applied rewrites 90.3%

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

   if 0.999999799999999994 < (/.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 86.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.2:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999998:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.03:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-19}:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \mathbf{elif}\;t\_2 \leq 0.9999998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx)))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_2 -0.03)
     t_1
     (if (<= t_2 1e-19)
       (/ -1.0 (* (/ (sin kx) (sin th)) (/ -1.0 (sin ky))))
       (if (<= t_2 0.9999998) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (-th * sin(ky)) * (-1.0 / 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 <= -0.03) {
		tmp = t_1;
	} else if (t_2 <= 1e-19) {
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	} else if (t_2 <= 0.9999998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (-th * Math.sin(ky)) * (-1.0 / 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 <= -0.03) {
		tmp = t_1;
	} else if (t_2 <= 1e-19) {
		tmp = -1.0 / ((Math.sin(kx) / Math.sin(th)) * (-1.0 / Math.sin(ky)));
	} else if (t_2 <= 0.9999998) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (-th * math.sin(ky)) * (-1.0 / 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 <= -0.03:
		tmp = t_1
	elif t_2 <= 1e-19:
		tmp = -1.0 / ((math.sin(kx) / math.sin(th)) * (-1.0 / math.sin(ky)))
	elif t_2 <= 0.9999998:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / 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 <= -0.03)
		tmp = t_1;
	elseif (t_2 <= 1e-19)
		tmp = Float64(-1.0 / Float64(Float64(sin(kx) / sin(th)) * Float64(-1.0 / sin(ky))));
	elseif (t_2 <= 0.9999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.03)
		tmp = t_1;
	elseif (t_2 <= 1e-19)
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	elseif (t_2 <= 0.9999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $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, -0.03], t$95$1, If[LessEqual[t$95$2, 1e-19], N[(-1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999998], 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))))) < -0.029999999999999999 or 9.9999999999999998e-20 < (/.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))))) < 0.999999799999999994

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 92.3

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 43.3

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

   7. Applied rewrites 43.3%

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

   if -0.029999999999999999 < (/.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))))) < 9.9999999999999998e-20

   1. Initial program 98.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 98.0

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 54.4

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

   7. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. frac-2neg N/A

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

       4. associate-/l/ N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. distribute-neg-frac2 N/A

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

       8. lift-/.f64 N/A

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

   11. Applied rewrites 53.4%

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

   if 0.999999799999999994 < (/.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 86.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.03:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-19}:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999998:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left({th}^{3}, -0.16666666666666666, th\right) \cdot \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + kx \cdot kx}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -1.0)
     (*
      (fma (pow th 3.0) -0.16666666666666666 th)
      (/ (sin ky) (sqrt (+ (- 0.5 (* (cos (* 2.0 ky)) 0.5)) (* kx kx)))))
     (if (<= t_1 5e-108)
       (/ -1.0 (* (/ (sin kx) (sin th)) (/ -1.0 (sin ky))))
       (if (<= t_1 2e-8)
         (*
          (/ (sin ky) (sqrt (+ (* ky ky) (- 0.5 (* 0.5 (cos (* 2.0 kx)))))))
          (sin th))
         (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 <= -1.0) {
		tmp = fma(pow(th, 3.0), -0.16666666666666666, th) * (sin(ky) / sqrt(((0.5 - (cos((2.0 * ky)) * 0.5)) + (kx * kx))));
	} else if (t_1 <= 5e-108) {
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	} else if (t_1 <= 2e-8) {
		tmp = (sin(ky) / sqrt(((ky * ky) + (0.5 - (0.5 * cos((2.0 * kx))))))) * sin(th);
	} 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 <= -1.0)
		tmp = Float64(fma((th ^ 3.0), -0.16666666666666666, th) * Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5)) + Float64(kx * kx)))));
	elseif (t_1 <= 5e-108)
		tmp = Float64(-1.0 / Float64(Float64(sin(kx) / sin(th)) * Float64(-1.0 / sin(ky))));
	elseif (t_1 <= 2e-8)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx))))))) * sin(th));
	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, -1.0], N[(N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-108], N[(-1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 82.2%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(\color{blue}{{th}^{3}}, \frac{-1}{6}, th\right) \]

      9. lower-pow.f64 35.8

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(\color{blue}{{th}^{3}}, -0.16666666666666666, th\right) \]

   8. Applied rewrites 35.8%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left({th}^{3}, -0.16666666666666666, th\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      9. count-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      11. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \mathsf{fma}\left({th}^{3}, -0.16666666666666666, th\right) \]

   10. Applied rewrites 30.8%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \mathsf{fma}\left({th}^{3}, -0.16666666666666666, th\right) \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-108

   1. Initial program 98.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 98.3

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

      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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 44.2

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

   7. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. frac-2neg N/A

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

       4. associate-/l/ N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. distribute-neg-frac2 N/A

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

       8. lift-/.f64 N/A

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

   11. Applied rewrites 43.4%

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

   if 5e-108 < (/.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))))) < 2e-8

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if 2e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.1

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

   5. Applied rewrites 67.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\mathsf{fma}\left({th}^{3}, -0.16666666666666666, th\right) \cdot \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.3)
   (/ -1.0 (* (/ (sin kx) (sin th)) (/ -1.0 (sin ky))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.3) {
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.3d0) then
        tmp = (-1.0d0) / ((sin(kx) / sin(th)) * ((-1.0d0) / sin(ky)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.3) {
		tmp = -1.0 / ((Math.sin(kx) / Math.sin(th)) * (-1.0 / Math.sin(ky)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.3:
		tmp = -1.0 / ((math.sin(kx) / math.sin(th)) * (-1.0 / math.sin(ky)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = Float64(-1.0 / Float64(Float64(sin(kx) / sin(th)) * Float64(-1.0 / sin(ky))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.3], N[(-1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sin[ky], $MachinePrecision]), $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))))) < 0.299999999999999989

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 93.9

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

      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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 33.3

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

   7. Applied rewrites 33.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. frac-2neg N/A

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

       4. associate-/l/ N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. distribute-neg-frac2 N/A

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

       8. lift-/.f64 N/A

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

   11. Applied rewrites 32.8%

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

   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 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.3)
   (/ 1.0 (/ (/ (sin kx) (sin th)) (sin ky)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.3) {
		tmp = 1.0 / ((sin(kx) / sin(th)) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.3d0) then
        tmp = 1.0d0 / ((sin(kx) / sin(th)) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.3) {
		tmp = 1.0 / ((Math.sin(kx) / Math.sin(th)) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.3:
		tmp = 1.0 / ((math.sin(kx) / math.sin(th)) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = Float64(1.0 / Float64(Float64(sin(kx) / sin(th)) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = 1.0 / ((sin(kx) / sin(th)) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.3], N[(1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $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))))) < 0.299999999999999989

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 93.9

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

      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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 33.3

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

   7. Applied rewrites 33.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 32.8%

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

   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 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.3)
   (* (/ (sin th) (sin kx)) (sin ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.3) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.3d0) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.3) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.3:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.3)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.3], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.299999999999999989

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 93.9

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

      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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 33.3

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

   7. Applied rewrites 33.3%

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

   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 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.3% 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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.3:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-12}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-12)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-12) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-12) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-12) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-12:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-12)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-12)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999998e-13

   1. Initial program 93.7%

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

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

      7. lower-/.f64 93.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 32.4

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

   7. Applied rewrites 32.4%

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

   if 9.9999999999999998e-13 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.1

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

   5. Applied rewrites 67.1%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-12}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-12}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-12)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-12) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-12) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-12) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-12:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-12)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-12)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999998e-13

   1. Initial program 93.7%

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

   3. Taylor expanded in 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 32.9

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

   5. Applied rewrites 32.9%

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

   if 9.9999999999999998e-13 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.1

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

   5. Applied rewrites 67.1%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-12}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-175}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + kx \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 4e-303)
   (*
    (/ (sin ky) (sqrt (+ (* ky ky) (- 0.5 (* 0.5 (cos (* 2.0 kx)))))))
    (sin th))
   (if (<= (sin kx) 1e-175)
     (sin th)
     (if (<= (sin kx) 2e-33)
       (* (/ (sin ky) (sqrt (+ (* ky ky) (* kx kx)))) (sin th))
       (/ -1.0 (* (/ (sin kx) (sin th)) (/ -1.0 (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 4e-303) {
		tmp = (sin(ky) / sqrt(((ky * ky) + (0.5 - (0.5 * cos((2.0 * kx))))))) * sin(th);
	} else if (sin(kx) <= 1e-175) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-33) {
		tmp = (sin(ky) / sqrt(((ky * ky) + (kx * kx)))) * sin(th);
	} else {
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= 4d-303) then
        tmp = (sin(ky) / sqrt(((ky * ky) + (0.5d0 - (0.5d0 * cos((2.0d0 * kx))))))) * sin(th)
    else if (sin(kx) <= 1d-175) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-33) then
        tmp = (sin(ky) / sqrt(((ky * ky) + (kx * kx)))) * sin(th)
    else
        tmp = (-1.0d0) / ((sin(kx) / sin(th)) * ((-1.0d0) / sin(ky)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= 4e-303) {
		tmp = (Math.sin(ky) / Math.sqrt(((ky * ky) + (0.5 - (0.5 * Math.cos((2.0 * kx))))))) * Math.sin(th);
	} else if (Math.sin(kx) <= 1e-175) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-33) {
		tmp = (Math.sin(ky) / Math.sqrt(((ky * ky) + (kx * kx)))) * Math.sin(th);
	} else {
		tmp = -1.0 / ((Math.sin(kx) / Math.sin(th)) * (-1.0 / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= 4e-303:
		tmp = (math.sin(ky) / math.sqrt(((ky * ky) + (0.5 - (0.5 * math.cos((2.0 * kx))))))) * math.sin(th)
	elif math.sin(kx) <= 1e-175:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-33:
		tmp = (math.sin(ky) / math.sqrt(((ky * ky) + (kx * kx)))) * math.sin(th)
	else:
		tmp = -1.0 / ((math.sin(kx) / math.sin(th)) * (-1.0 / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 4e-303)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx))))))) * sin(th));
	elseif (sin(kx) <= 1e-175)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-33)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + Float64(kx * kx)))) * sin(th));
	else
		tmp = Float64(-1.0 / Float64(Float64(sin(kx) / sin(th)) * Float64(-1.0 / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= 4e-303)
		tmp = (sin(ky) / sqrt(((ky * ky) + (0.5 - (0.5 * cos((2.0 * kx))))))) * sin(th);
	elseif (sin(kx) <= 1e-175)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-33)
		tmp = (sin(ky) / sqrt(((ky * ky) + (kx * kx)))) * sin(th);
	else
		tmp = -1.0 / ((sin(kx) / sin(th)) * (-1.0 / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-303], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-175], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-33], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < 3.99999999999999972e-303

   1. Initial program 93.7%

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

   3. Taylor expanded in 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 48.3

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

   5. Applied rewrites 48.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 37.3

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

   7. Applied rewrites 37.3%

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

   if 3.99999999999999972e-303 < (sin.f64 kx) < 1e-175

   1. Initial program 72.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 54.7

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

   5. Applied rewrites 54.7%

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

   if 1e-175 < (sin.f64 kx) < 2.0000000000000001e-33

   1. Initial program 97.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 2.0000000000000001e-33 < (sin.f64 kx)

   1. Initial program 99.3%

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

      1. lift-*.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 99.5

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

      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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 51.4

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

   7. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. frac-2neg N/A

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

       4. associate-/l/ N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. distribute-neg-frac2 N/A

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

       8. lift-/.f64 N/A

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

   11. Applied rewrites 50.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{-1}{\sin ky} \cdot \frac{\sin kx}{-\sin th}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq 4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-175}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + kx \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sin kx}{\sin th} \cdot \frac{-1}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1.95 \cdot 10^{-60}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
      1.95e-60)
   (* -0.16666666666666666 (pow th 3.0))
   (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)))) <= 1.95e-60) {
		tmp = -0.16666666666666666 * pow(th, 3.0);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1.95d-60) then
        tmp = (-0.16666666666666666d0) * (th ** 3.0d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1.95e-60) {
		tmp = -0.16666666666666666 * Math.pow(th, 3.0);
	} 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)))) <= 1.95e-60:
		tmp = -0.16666666666666666 * math.pow(th, 3.0)
	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)))) <= 1.95e-60)
		tmp = Float64(-0.16666666666666666 * (th ^ 3.0));
	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)))) <= 1.95e-60)
		tmp = -0.16666666666666666 * (th ^ 3.0);
	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], 1.95e-60], N[(-0.16666666666666666 * N[Power[th, 3.0], $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))))) < 1.9500000000000001e-60

   1. Initial program 93.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.6

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

   5. Applied rewrites 3.6%

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

   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 3.4%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 15.0%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 1.9500000000000001e-60 < (/.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 92.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.0

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

   5. Applied rewrites 62.0%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1.95 \cdot 10^{-60}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot t\_1\\ \mathbf{elif}\;\sin ky \leq 10^{-8}:\\ \;\;\;\;\left(\left(-ky\right) \cdot t\_1\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ -1.0 (hypot (sin ky) (sin kx)))))
   (if (<= (sin ky) -0.02)
     (* (* (- th) (sin ky)) t_1)
     (if (<= (sin ky) 1e-8) (* (* (- ky) t_1) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = -1.0 / hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (-th * sin(ky)) * t_1;
	} else if (sin(ky) <= 1e-8) {
		tmp = (-ky * t_1) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -1.0 / Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = (-th * Math.sin(ky)) * t_1;
	} else if (Math.sin(ky) <= 1e-8) {
		tmp = (-ky * t_1) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -1.0 / math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = (-th * math.sin(ky)) * t_1
	elif math.sin(ky) <= 1e-8:
		tmp = (-ky * t_1) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * t_1);
	elseif (sin(ky) <= 1e-8)
		tmp = Float64(Float64(Float64(-ky) * t_1) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -1.0 / hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = (-th * sin(ky)) * t_1;
	elseif (sin(ky) <= 1e-8)
		tmp = (-ky * t_1) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-8], N[(N[((-ky) * t$95$1), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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. 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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.3

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 43.8

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

   7. Applied rewrites 43.8%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 1e-8

   1. Initial program 85.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.8

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

   7. Applied rewrites 97.8%

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

   if 1e-8 < (sin.f64 ky)

   1. Initial program 99.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 58.9

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

   5. Applied rewrites 58.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 10^{-8}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 6.5e-15)
   (* (* (- ky) (/ -1.0 (hypot (sin ky) (sin kx)))) (sin th))
   (*
    (/
     (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 <= 6.5e-15) {
		tmp = (-ky * (-1.0 / hypot(sin(ky), sin(kx)))) * sin(th);
	} 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 <= 6.5e-15)
		tmp = Float64(Float64(Float64(-ky) * Float64(-1.0 / hypot(sin(ky), sin(kx)))) * sin(th));
	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, 6.5e-15], N[(N[((-ky) * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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 < 6.49999999999999991e-15

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.3

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

   7. Applied rewrites 64.3%

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

   if 6.49999999999999991e-15 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 23.8% 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 92.8%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 29.8

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

5. Applied rewrites 29.8%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 18: 13.2% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (fma (* -0.16666666666666666 (* th th)) th th))

C

double code(double kx, double ky, double th) {
	return fma((-0.16666666666666666 * (th * th)), th, th);
}

Julia

function code(kx, ky, th)
	return fma(Float64(-0.16666666666666666 * Float64(th * th)), th, th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] * th + th), $MachinePrecision]

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 29.8

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

5. Applied rewrites 29.8%

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

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 16.1%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 16.1%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(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)))